It looks to me like
Federation planets mainly depend on a
barter system. The
replicator and
Transporter systems leveled a lot of trade imbalance, but they don’t run on thin air.
Technology,
raw materials,
power supplies, services, and
information still have
value.
Ships still need dilithium for their warp cores. Replicators won’t work without deuterium. And where does all that food come from, anyway? It’s converted from nutritional supplements... surely not appetizing, but it’s got to come from somewhere.
And though we’ve seen holograms and androids performing in many service positions, I can’t believe the technology is available to everyone... and even if it is, who’s maintaining it?
Also, the Federation in Star Trek does have currency. In the TOS episode, "The Trouble with Tribbles," the character Cyrano Jones sold tribbles for Federation credits. And in the DS9 episode, "Past Tense, Part One," we actually see a credit chip.
Source: Star Trek Encyclopedia