One way to avoid having one crew would be to follow more than one ship. The show could be about a fleet (in the rather loose terms for a non-war Trek), or about an event or period of time. It could have an episode, a mini-arc, or even a whole season (depending on length of season) about one ship, then switch to another. Depending on what the overall show was about, it could even follow a non-Starfleet ship for a story or two. It could (and probably would) return to past ships, to see what they are up to now.