There was definitely a money system in the Star Trek Universe. It was a credit based system (heck even the monetary unit was called a Federation Credit).
You especially saw this in the DS-9 series where it played a more prominent role (as well as the Ferengi) in the store. Even today we are moving to this type of system with debit and credit cards, although cash is still a valid form of currency. Also, you should keep in mind that most of these series where set on Military vessels where there is much less need to have money at all.
Even though they were in a time of post scarcity there was still uses for money - which are outlined in the above mentioned wikipedia article.
TheThese uses boil down to:
- A bartering tool between the United Federation of Planets and other governments
- A means of internal budget allocation in the United Federation of Planets
- A way for Federation citizens to barter for objects that cannot be replicated