How I've always seen it:

Starfleet (and perhaps a large portion of the UFP) do not use money or valuable materials to pay their workers and other organisations within Starfleet and likely those other UFP worlds under the same philosophy.

However, they do need to deal with many civilisations that do not subscribe to the same ideals. As such, they would likely maintain funds by trading to and from these civilisations.

It would be possible as others suggest to either bill Starfleet directly, or for officers to be able to requisition various precious items/currency as/when needed.