I'm going to recommend a system I don't have any direct experience with, so take the following with a grain of salt (or a small dilithium crystal...).
I'd use FATE or one of the FATE-based systems ie Spirit of the Century, Starblazer Adventures.
They won't give you rich tactical experiences, but you'll have plenty of support for playing through/resolving the kind of situations often found in Star Trek episodes. For example, the way the crew defeat the Kelvan in "By Any Other Name" -- by Kirk using his way with the ladies, Scotty using his way with booze, and McCoy using injected drugs-- would work wonderfully under FATE.
The big question is: do you want to play a science fiction campaign set in the Star Trek universe, or do you want to play a campaign that emulates the actual episodes?
I'd use FATE or one of the FATE-based systems ie Spirit of the Century, Starblazer Adventures.
They won't give you rich tactical experiences, but you'll have plenty of support for playing through/resolving the kind of situations often found in Star Trek episodes. For example, the way the crew defeat the Kelvan in "By Any Other Name" -- by Kirk using his way with the ladies, Scotty using his way with booze, and McCoy using injected drugs-- would work wonderfully under FATE.
The big question is: do you want to play a science fiction campaign set in the Star Trek universe, or do you want to play a campaign that emulates the actual episodes?