I'm a little surprised no-one has discussed this since the start of "Enterprise".
Suppose you wanted to run a game concurrent with the TV show "Enterprise". How would you do it, all filled with fear of contradicting the show? I remember FASA's first stab at ST:TNG around the end of season 1 of TNG - a lot of wild speculation... I'm not actually considering such a game, but it does present an interesting exercise in my opinion. (Though heck, if there's some really funky ideas, I'm flighty enough to consider one...)
Personally, I think you'd almost have to do a non-political, exploration game, far away from major civilizations. As for known Trek aliens, I think you'd mainly be using humans, Vulcans, Klingons, and Andorians - any others may be a bit tricky since "Enterprise" may show their first contact.
The ship itself presents a difficulty - do you make it like Enterprise and hope the show doesn't reveal that she's one of a kind. Do you make your own class of ship and give it a funky hull registry - CA-47?