There's a new "Star Trek" RPG coming out. I'm probably going to skip it, for various reasons, not least that I'm kinda hoping to play, rather than run. However, hypothetically speaking, if I were to run a "Star Trek" campaign, what would I do?
Era of Play
I'd set the campaign in the era of the Original Series movies. In fact, to be more specific, I'd set it in 2295 - two years after the first section of "Generations", and so two years after the end of that movies series.
The Ship and the Crew
The reason for that specific year is that it would allow me to set the campaign aboard the Enterprise-B without also being laden down with lots of fixed lore to work around - that ship only has two (or three) named crew-members, any of whom could have been re-assigned in the meantime, but it also has the famous name.
The players would be free to create pretty much any suitable character for the campaign. In the event that they chose to play the captain or helmsman, then the characters from "Generations" would be re-assigned; if not, those characters would remain as NPCs. (In general, my preference for a ST-style game would indeed be for the captain to be an NPC, with the bulk of his/her decisions made by the consensus of the players, thus allowing the players to drive the campaign without having a pesky chain of command to worry about.)
Campaign Structure
The campaign would see the Enterprise out in deep space on another Five Year Mission, at least initially. In keeping with the series, it would be mostly episodic, maybe with some arcs being added to the campaign later. (Indeed, there's an argument for using the game as another "open tabletop", with many players each bringing their own crew-member to the table, and with the selection of players determining which characters are in focus in the session.)
Perhaps importantly, because the ship will be "out there", they'll be in a position where (1) they can't rely on the might of the Federation for backup but will have to deal with problems themselves, but also (2) they won't necessarily be constrained by the Federation's rules all the time - sure, there's the Prime Directive, but if they break it, who's to know?
Other Stuff
Star Trek has always been fairly optimistic, and when it has gone wrong that has tended to be when it has gone too dark. Even when facing off against world-ending threats, the feeling has tended to be quite upbeat. Heck, even "Voyager", where the crew faced the prospect of living their entire lives on the ship, remained optimistic. So I'd certainly intend to maintain that feeling in the campaign.
Equally, though, Star Trek has tended to draw quite heavily from the zeitgeist - in particular, an awful lot of the Federation/Klingon interaction in TOS was a parallel for the Cold War. Of course, that tension largely came to an end in "The Undiscovered Country". Consequently, I'd be inclined to draw from the spirit of our times, where the main threats tend to be smaller-scale and also not really amenable to being solved with big guns. Plus, I'd be inclined to have the Federation (and the crew of the Enterprise) face some measure of existentialist angst - they're in the most advanced ship in the fleet, a heavily-armed warship... and now the threat that the ship was designed to counter has been largely removed. They find themselves on a scientific and exploration mission that the Excelsior-class isn't really designed for. So, what are they for?
Oh, and of course the powers-that-be will have insisted on installing a plaque in deflector control, which has turned out to be a spectacular mis-step. Especially since everyone wears red shirts in the movie universe...