I always thought that the stories in Star Trek were almost universal. The thing that made them different was Star Trek's take on human potential and very optimistic. Star Trek, I thought, was always trying to show humans at their best, rising to the occasion and overcoming the challenge put before them. The different races tended to show distinct facets of humanity, this was even the case with the TOS crew and especially of the three main characters. Spock was all Logic, McCoy was humanity and emotion, and Kirk was the focal point, a mixture of both. So if you Babylonize your campaign too much, it stops being Trek and starts being B5 in different uniforms.
We like to keep as close to what we see on TV and make it our own campaign at the same time. Starfleet doesn't go rogue except for some extreme cases. Watch "The Wounded", and Starfleet is very efficient thanks to certain races that are just good at that, but there is the occasional boondoggle, and that's were the stories get interesting, and occasionally there's some shooting. We set our game in the 2340's, so the ship was a Constellation-class and later an Ambassador-class. This time frame is a little fuzzy so I had a lot of room to experiment with storylines, my players ended up with some battles in the Cardassian wars and skirmishes with the Tholians, oh what fun.
A couple of them didn't even like the TOS, or hadn't seen much of it, but they were big Star Wars fans. I just had to explain to them that Trek was a little different. Now they're both big TNG fans and like the movies.
Boy I'm really starting to ramble and babble.
Well, to sum up (I think), just make the stories interesting and keep it fun and it won't matter if the players are Trekkers or not.
"Retreat?! Hell, we just got here!", annonymous American Marine, WWI
"Gravity is a harsh mistress....", The Tick