You could bring in a character like Mr. Mudd. He was always doing something not quite legal and always got out of it until he runs into the PC ship. Maybe he's running from some beings who want to really "Punish" him, and he let's himself get caught by the Starfleet ship so that they will protect him.
You could let the crew have shoreleave on a new world and get caught up in an honor duel to the death.
But some rules to follow.
1. Somebody in a red shirt is going to have a BAD day.
2. Somebody is going to loose a shirt, this is usually a guy, but hey, let's be equal, a lady could loose her dress and all of the guys have to turn around.... or hand out snacks and pull up chairs.
3. Someone has to say, "I'm a (PC's job), not a (whatever)!"
4. Even with phasers, swords, and pocket nukes, every fight degenerates into a barroom brawl.
"Retreat?! Hell, we just got here!", annonymous American Marine, WWI
"Gravity is a harsh mistress....", The Tick