My campaign was set on a mothballed-for-twenty-years Ambassador class with an Andorian captain who had lost his last ship to a Romulan ambush, and a crew scraped together from crewmen that all the other captains in Starfleet wanted off their own ships. Make them the underdogs, give them a chance to get to know each other and see how they can work together. Then throw them into the brown sticky stuff and watch them fight their way out of it.
I find this kind of campaign much more fun, interesting, and believable than TNG's perfectly happy respectable well-adjusted crew. It also seems to fit most players better than a typical plain-vanilla crew. After all, it's fun to thumb your nose at the establishment once in a while.