While there's merit in discussing whether the Federation has some sort of a money economy, I think it's pretty obvious that a pure, American-style money economy is not something we've seen on Star Trek. People are quick to point to Crusher straight-up buying that cloth at Farpoint Station, but that was decidedly outside the borders of the Federation. Other than that, we've never seen our characters actually buying anything with money within the borders of the Federation. Babylon 5 had credit chits and reading slots on every vendor's counter, so it should be a pretty easy thing to film (e.g it's not like they just didn't feel it was necessary to show the characters checking out), but we haven't seen anything like this in the Federation.
I personally think the Federation has some sort of a hybrid economy; there is some sort of money, but there's not much you could actually spend it on in the civilized core of the Federation. When you have replicators, you can't really talk of scarcity. We've never been told what kind of raw materials the replicators use, but I'd guess it's hydrogen, e.g. the most abundant element in the entire universe. It's practically impossible to run out of hydrogen, and I don't really see the Federation hoarding it just so that they could generate artificial scarcity.
And most important of all, if one were to propose that the Federation has a pure money economy, they have to contend with the fact that this presupposes that there are haves and have-nots in the Federation. There are people who have to subsist on replicator rations because they can't buy more replicator material. There are people who have to rent a room in a dilapidated concrete apartment block with five other people because they are stuck at a low-paying job and can't even afford to rent their own place. That there are people who are forced to sell their bodies or work for the mafia to sustain themselves. Not on the frontier that the Federation can't police, on Earth. Which is supposed to be a paradise. I can't say this meshes well with what we know about the Federation.
Simply based on what we've seen about Earth, it's pretty obvious that at the very least there's an extensive welfare state, with the government providing most basic amenities like food, housing, healthcare, education or public transport. Yeah, Raffi lives in a cabin. But it's pretty obvious it was her personal choice. Yeah, Picard pays Rios for the transport. But that doesn't mean you have to pay everyone. You pay those who can fly off the books. But a Federation-operated starliner to Betazed? That's probably free of charge.