How well-known would each crew be to the public?

Suffice it to say, I don't think Picard would enjoy being famous.

Interviewer: good evening ladies, gentlemen, other genders and species. Tonight in 'the Federation's Finest' we have two very special guests, the famed captain of the Enterprise, captain Picard and one of his trusty aides, Lt. Commander Data! Show them your appreciation!
Interviewer: So, Jean-Luc, I can call you Jean-Luc, right?
"Jean-Luc" Well, erm, I'm not quite comf...
Interviewer: So, Jean-Luc, what's it like to be the captain of the Federation flagship, and commanding the Starfleet very best and brightest and taking all decisions independently in many cases? Don't you feel like a God out there when you decide whether a primitive race lives or dies, or when to lobby a few photon torpedoes against a Klingon or Romulan warbird?
Capt. Picard: Well, first of all you have to understand there are moral principles and guidelines we have sworn to keep, the ideals of the Fede...
Interviewer: Yes, yes, that's all very interesting and all, but how does it feel to be assimilated or kill your own people at Wolf 359?
Capt. Picard: That's something I'd rather not discuss as it brings up all kinds of unpleasant memories and emotions.
Interviewer: Come on Jean Luc! You know what it feels like, and we want to know! Don't hold back!
Capt. Picard: I'd really rather not ...
Interviewer: Do you gradually start to hear the voice of the collective? Do you feel control of your body slowly slipping away or is the takeover instantly? Do you feel any pain when it happens? Is it still possible to resist at first? Do you still have feelings of remorse regarding those 11,000 people you murdered? Can you sleep at night knowing what you did? What's it like firing weapons against ships you know your friends are on?
<Captain Picard looks very uncomfortable>
Data: Captain, if I may? Remote readings of his stress hormone levels and after his capture suggested that he was under extreme duress during the entire experience, and medical evaluations showed he needed a extensive recovery and significant psychological trauma, even when he tried to act as if nothing happened and
<capt. Picard hissing> Shut up, Data!
Data. Yes, sir.
Interviewer <sensing the awkward moment>: we'll be right back after a few words from our obviously non-commercial sponsor, we're not in the Ferengi Alliance after all, but in the Federation! ha! ha!
<after the break>
Interviewer: Well, Jean-Luc, see that door? Behind that door is our first mystery guest! Someone you know. I'll give you a few hints and let's see if you can guess who it is! OK?
capt. Picard: Erm ...
Interviewer: Your first clue: you know her very intimately! Ha! ha!
capt. Picard: I wouldn't know anyone that fits that description. I have no room for a wife or partner.
<Whispering urgently to Data> Are we absolutely certain Beverly went to that Eroticon V conference and that she cannot be here? Probability analysis, mr. Data!
Interviewer: Time's up! Your second clue, Jean-Luc! Try to keep up! The ship's gossip channels have it she seduced you with a piano in a Jeffries' tube!
<Picard blushes when he realises his affair wasn't as private as he thought it was>
Interviewer: I see you guessed it, Jean-Luc! Open the door!
Perhaps things used to be named after Kirk, only to be renamed when somebody took issue.

As for Pike, maybe his birthday is significant in Starfleet and space studies circles. The kind of thing others learn from social media and Google doodles.
Just because we haven’t seen anything named after Kirk doesn’t mean something doesn’t exist. All it takes is a writer on Lower Decks, Prodigy or Academy to mention the “James T. Kirk Command School”, the “statue of James T. Kirk at Starfleet HQ”, the “UFP James T. Kirk Medal of Honor” (Pike I think has the Medal of Valor), or “Captain, we already have the James Kirk on its way to provide assistance can you respond as well”

And I would expect Bajor has tons of stuff named after Sisko either by name or as “The Emissary”.

Canon is wide open on this kinda stuff.
Kirk would have a shuttle aboard any of the Enterprises in service named for him after his first "death". Same for Decker Sr. aboard any Constellation in service after his own death.
In "real life", people who are well-known over any period of time tend either to be the "first" of something (Armstrong and Aldrin) or a major leader during a period of change or considerable stress (Genghis Khan, Stalin). There's a third group who might be called legends - there's a basis in fact but the things people know about them are frequently untrue or highly exaggerated (Paul Revere, Dick Turpin)

That something is called after someone is not, in itself, a sign that the individual is well-known among the general populace. I live in a road that is called after a very wealthy, very influential newspaper magnate...of the 1930s. I doubt very much that the current denizens even know the road was called after a person, never mind for what they were known. It's not uncommon for, say, university or business, buildings to be called after someone significant to an organisation but who is unknown outside.

Looking at Star Trek, Archer and possibly his senior crew would qualify as "firsts" especially as Archer also becomes a significant political leader. After that, all our crews are "just" another spaceship crew who aren't doing anything that would be seen by the sapient-in-the-street as being striking or significant. Star Fleet would be the people who decided what "news" or information is released: the general public would be completely unaware of what went on. Even the whale probe situation which clearly radically affects the whole Earth population is likely to end up associated with the senior staff giving the orders on Earth.

Things like the Pike medal commemorate events significant inside Star Fleet. Most people who save lives at the risk of their own remain generally unknown or their names forgotten within days.

I suspect DS9 might be known as the space station that found the wormhole/ is by the wormhole - but the names of the staff would not be known (think of the staff of the ISS: people have heard of the ISS but few could tell you who was there).

TL;DR: The general public are unlikely to know anyone except for, perhaps, senior officers of the NX-01

Edit to correct typo
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That's the way fame goes, though. Quite a few people have heard of Columbus and Magellan and De Gama and Cook and so on. Expedition commanders and ship's captains. How many know offhand the names of the two of Magellan's crew who actually made it all the way around the world? Or any of their lieutenants?
I was rewatching TNG Relics the other day, and something that I always found interesting was that despite Scotty being very well known to audiences, the crew of the Enterprise-D treat him as they would any other guest of the week. It got me wondering -- how well known would Scotty have been to people living in the 24th century, both in and out of Starfleet? How well would any of them have been known? Generals and admirals in real life are known to history buffs, but the general public can only name the most famous, if that. I ran down a list of each major character's best known deeds and I'm curious if people agree. This knowledge is only based on what we see on screen.

Kirk - Kirk's Enterprise has been shown to have quite the reputation even a century later within Starfleet, however I imagine the general (Federation) public would be largely unaware of his five year mission. I think he would be best known for his stopping the Whale Probe first and foremost, followed by his involvement in the creation of the Genesis planet and barging into the Klingon peace talks at Khitomer.
Spock - Closely associated with Kirk's legacy, I think he would be equally known to later generations as the ambassador to a major galactic power, as well as his (assumed) death during the Romulan supernova.
The rest of the TOS cast - Again, closely associated with Kirk. Sulu might have a slight edge due to being captain during the Praxis incident, but the others would be largely out of public view.

Picard - By far best known for being assimilated by the Borg and leading the disaster at Wolf 359. Like Kirk, certain missions of the Enterprise-D would be well-known within Starfleet history, and I have to imagine his recounting of the events of The Inner Light and The Chase would have made great splashes in Archaeology circles, but those are smaller groups. It's a bit muddled how much of First Contact would truly have been made public, but it probably would've helped to redeem his image, as well as the events of Picard season 3. Romulan citizens would also probably know him well due to the events of Nemesis, and maybe Klingons for his role in making Gowron chancellor.
Data - Data has been shown to be well-known to other Starfleet officers (Bashir and B'elanna) due to being the first Soong-type android, and I imagine that would extend somewhat to the public. Aside from that, I don't think he would have any deeds that would make him more famous than his Enterprise-D coworkers though.
Worf - Probably fairly well-known in Starfleet for being the first Klingon in the service, as well as his postings on the Enterprise and DS9. However, while Federation citizens might not know him, Klingon ones certainly would. He would have been a footnote on the attack on Khitomer, but famous for slaying Duras and Gowron, as well as being made Klingon ambassador to the Federation under Martok (although that didn't seem to last too long).
Riker - Based only on what we see, Riker would've been well-known for being the captain who stopped the Borg attack following Wolf 359, but would largely play second fiddle to Picard.
The rest of the TNG cast - Maybe Geordi had a great and famous career later on, but solely from what we see on screen, they would've remained largely out of the public eye.

Sisko - Sisko is a household name on Bajor from the beginning of DS9, and I think he would have been very well-known in Starfleet as the commander of a vital strategic outpost, but that might be it. His most famous actions would probably be the mining of the wormhole and subsequent recapture of DS9, and the signing of the peace treaty on DS9, so military buffs would know him well. Who knows how Federation news reported that business with Dukat right as the war ended.
Odo - As the only Founder on this side of the war, Starfleet brass would've kept a close eye on Odo, and word of his existence would certainly have slipped out as the war raged on. It's possible he was the best known of the DS9 crew, but it's hard to say.
Kira - Probably fairly famous on Bajor due to being the Bajoran liaison to DS9. The Bajoran tabloids would've had a field day with her though.
O'Brien - The most important man who ever lived. Perhaps not the most famous though, he stayed largely out of the limelight.
The rest of the cast - Well-known to history buffs, but I think largely unknown to the average citizen.

Voyager as a whole would have been a historical footnote until the midpoint in the series when they re-established contact with the Federation. Following that, they would've been of strategic interest to Starfleet, and possibly fairly well-known publicly, but I think it wasn't until their return to Earth where they became really famous. They're also a rare example of a crew that would've become more well-known as history continued, due to the Federation expanding into the Delta Quadrant and bringing more and more of their adventures to light. Of the crew, Janeway as the captain would've gained the largest recognition, while Seven, as an ex-Borg, probably would've been second (although that certainly didn't help her career). Seven's later involvement with the events of Picard season 3 also would've bolstered her image.
I did struggle to find an event where a single member of the crew would've received more acknowledgement than the others upon their return, with the exception of Threshold. Aside from being trapped in the Delta Quadrant, Janeway and Paris would've been noted as the first to break the Warp 10 barrier, putting them alongside real figures like Chuck Yeager.

Archer - I think of everyone on this list, Archer probably would be the most famous to the public. He and the crew were shown to be pretty publicized at the beginning of the show, but their involvement in the Xindi Crisis would've made them a household name. Even shortly after we're told elementary schools were being renamed after him. His later involvement in the foundation of the Federation means he would've been in every history curriculum going forward.
The others - I don't think any of the others would've gotten the same level as Archer did individually, although they would've also gained attention for the Xindi Crisis. T'Pol broke a number of Vulcan records for staying on a starship, but those would likely have only been known as a curiosity to Vulcan officers. Trip, Phlox, and maybe Hoshi too might have gotten significant notoriety in their fields.

Michael Burnam - Probably best known to history as the instigator of the Federation-Klingon War. All of her future actions took place on the classified Discovery, whose existence was completely sealed by Starfleet in the 23rd century. After reappearing in the 31st century however, she would've gained quite a bit of attention in her actions to reunify Starfleet and solve The Burn, easily the best known galactic-wide event of the time. I'll admit to you I haven't seen Seasons 3 onward of Discovery, so I can't give better specifics.
Rest of the crew - Officially forgotten during the 23rd century, they would've been lost to time until their actions during the 31st. Saru as captain would've gained the most infamy during this time.

Lower Decks
Boimer - The namesake of the "Boimler Effect", something never forgotten by future generations.
The rest - They seem rather well-known... by other lower deckers of California-class ships. I guess that's something. Mariner also may have made a name for herself for the events of the end of Lower Decks season 4, but that probably didn't get exposed to the public either.

Strange New Worlds
Pike - He has been shown as the namesake for a number of medals of valor, implying that he has a very positive reputation around Starfleet, even during his life. However, I don't think he would be very well-known. The best evidence of that is that his best known event to audiences, that of his eventual fate, is completely classified in-universe.
Rest of the crew - Number One might be of legal significance in the future, but that's about it. I do get the impression the show is building up to something big for La'an, but we'll have to wait and see.

I've never seen Prodigy, however from what I know of it they are completely obscure characters to the larger galaxy.
It's funny you bring this up, because I always thought it was ridiculous sending senior officers from the NX-01 to infiltrate Terra Prime only months after the Xindi mission. It'd be like having Neil Armstrong and Buzz Aldrin going on some under cover mission after Apollo 11.

And, sure enough, they do, in fact, get caught.
That's the way fame goes, though. Quite a few people have heard of Columbus and Magellan and De Gama and Cook and so on. Expedition commanders and ship's captains. How many know offhand the names of the two of Magellan's crew who actually made it all the way around the world? Or any of their lieutenants?

True, though I also suspect peoples' specialties would come into play.

It's well possible that the average Federation citizen or perhaps even the average Starfleet ensign who didn't graduate in Engineering never heard of Scotty (or perhaps just as a passing and quickly forgotten reference when reading about Kirk's crew), whereas those that did the Engineering track frequently encountered his name in the textbooks they had to read.