The best game to use the Star Trek franchise name is Star Trek Online. This game is an MMO of PC based approximately thirty years after the events of the destruction of Romulus. War has broken out between the Federation and Klingons, so playable faction battlelines are drawn.
Unlike other Trek games released for the PC, this one is possibly the best representation of making a Trek game feel like trek instead of being some poor rip-off of Command and Conquer (Star Trek: Armada 1 & 2)
Given its nature of being an MMO, it is constantly under development and improvement. Most notably when "seasons" (much like the show) are released. However there are interim patches that occur. These mainly occur as bug fixes and can contain minor content updates/tweaks.
There is plenty of room for Role Playing, PVP and Crafting.
Recently the Foundry has come out of Beta. This facility enables players to be the directors of their own episodes by writing an episode with dialogue, specific events and tasks.