Actually, there is a conflict on this topic within the "original series", when extended to include the feature films. In several episodes of the old TV series, there are references to the use of money by people aboard ship. While on board a Space Station, the crew members on "shore leave" have to buy drinks. Uhura asks: "how much for the Tribble". People are heard wagering in monetary units which are simply called "credits". The "parallel universe" Kirk tries to bribe Spock.
However, when we fast-forward to Star Trek IV - The Voyage Home, there is at least one scene that leads the audience to think that money is not used in Kirk's time (the restaurant scene). That might be explained away as a cashless (as contrasted with moneyless) period. Kirk's 23rd-Century debit card would probably not work at Pizza Hut.