For starters, there'd almost certainly be no "Legacy" and no Sela.
I'm not sure what they would've had Worf do. They never DID say what his job in season 1 was, did they? It's possible they might've still had some of the Klingon-heavy episodes like "Sins of the Father" and "Redemption" (though again, no Sela).
They MIGHT have still done "Yesterday's Enterprise" (but probably with a very different final act).
Any other changes you can think of?
I would have been happy with no Sela. What a waste of the actress' time and effort (with a really horrible wig, to boot!).
Worf became The Character That Ate The Show - not only in TNG, but also DS9. I came to absolutely HATE him. I don't give a damn about Klingon politics, Klingon religion, all that "macho" swaggering around that apparently ALL Klingons are required to do - if Michael Dorn could play a Klingon lawyer in TUC, why couldn't he have played some other kind of Klingon character in TNG than the stereotypical fighter?
Well, I'm going to be a voice of dissent and say that Denise Crosby staying is one of the few things that could've really improved TNG, assuming the writers got a better grasp on her character in later seasons (which they almost surely would've, as they got better grasps on all the others). Her character had far more potential for development than Worf, who was always kind of one-dimensional and never seemed to really learn much, even through all seven seasons. Seeing Tasha struggle to balance her passionate, impulsive nature with the control and sophistication that made her so respect Picard would've been very interesting.
Worf could've replaced Geordi at the helm when he became chief engineer, thus ending the limbo. And frankly, I would much rather have seen episodes focusing on Tasha's difficult past and personal struggles than the Klingon episodes, which have always bored me.
Absolutely. The only thing about Worf that changed over the seven years was his haircut.
Tasha was the exception to the rule that the Federation Is Perfect. There is a FANTASTIC fanfic I'm reading about her Starfleet Academy years, during which she's not only doing her damndest to get through her classes, but also to deal with her personal demons - learning to trust people, develop social skills appropriate to Earth and Starfleet culture, and dealing with PTSD.
Most of the Enterprise crew either come from Earth or from highly developed, socially and economically stable colony worlds (or the alien equivalent). They can't possibly imagine what life is like for people who have no choice but to live as Tasha did when she was young. She could have contributed invaluable insights to Picard and the other crew on how people think and behave in such situations.
They probably only thought of Ishara as a way to mess with the crew's heads, since Tasha was dead. But they certainly could have done the character either way. They may have had a "Tasha goes home" type of story had she stayed.
That would have worked out even better. I hated it that Tasha wasn't around to defend herself against the nasty stuff Ishara kept saying.
I liked Tasha but it was also evident that nine characters were too much. Geordi and Worf had no decent position during the first season, a pilot rarely has much to say and Worf was often just an extra who stood around at the five consoles behind tactical.
Evidently the producers felt differently. DS9 had eight, later nine, main cast members (plus a dozen or so significant recurring ones), while Voyager had nine. Both had one cast member leave the show, but unlike TNG, they both introduced a new replacement character.
Admittedly, though, in both cases, characters fell by the wayside and were underused. (Jake, Harry)
Even on Enterprise, with only seven (would've been eight had it continued), a couple wound up being used less than could've been.
If they thought 8 or 9 were too many to write for, they should try soap writing, where there may be a couple of dozen characters.
They could've taken out the episodes involving Deanna's mother (because everyone hates those) and some of the boring Klingon stuff. Then they would've had plenty of time to utilize Tasha's character, and the show would've been better for it.
Some of the Lwaxana Troi stuff was pointless, but most of it was at least funny. The episode about the scientist who was expected to submit to euthanasia just because he'd reached mandatory life expectancy was a really excellent episode.
IMO, Worf was essentially meant to be the token Klingon on the Enterprise, there really just to show how much progress had been made since TOS.
Worf was supposedly the
first Klingon in Starfleet, if I remember right. Considering all the decades between the movies and TNG, why would he be the first? Shouldn't there have been any before then? Even as part of a cultural exchange?
I tend to think that if Tasha had stayed, they would have split tactical and security between them, with Tasha becoming the chief tactical officer (staying on the bridge firing the weapons), with Worf as the one doing the more physical stuff under her direction. As a result, I believe there would be episodes where Tasha's more of a presence than Worf and vice-versa.
This would have worked fine, as long as she got to do more than just fire weapons.
A part of me also thinks that Tasha and Geordi might have been a couple. While Worf would probably be the more obvious choice, I think for that very reason they wouldn't be (they may look at each other more as fellow warriors or even like siblings rather than as lovers). On the other hand, Geordi would be the least likely, and sometimes opposites do attract...
It's too bad they never explored that relationship. Geordi made it very clear, early on, that he was attracted to Tasha. I don't think she considered him as any more than a friend, but then that's how she related to all the crew.
Actually you could have still had the Yesterday's Enterprise plot (with Crosby) pretty much as it happen, with the exception of some of Guinan's dialog, and the tag scene in ten forward.. And the Sela storyline too. When Sela first appears on the main viewer, Tasha could have been at tactical with a very surprised look on her face.
How...?

You would have had to delete Guinan completely from the episode, as she's the one who kept insisting that Tasha was supposed to be dead. And there's no way that Tasha could go back in time, become Sela's mother, AND have stayed on the Enterprise in the present.
I don't think Tasha was an interesting enough character and Crosby a good enough actor for the writers to construct good plots around. She'd just fade into the background with the occasional bone tossed her way once or twice a season. Fans online would probable refer to her in the same "loving" terms they use for Travis and Harry.
Ahem. SOME of us like Tasha. In fact, she was my favorite TNG character.