Now I have seen one Star Trek movie, and six Star Wars movies. I didn't watch the TV series, or read any books. So I am basing everything on those 7 movies. You have to take away the force. With the force this is a no-brainer, Luke Skywalker wins easily. So if you take away the force you're left with two humans who traveled in time and space to defeat the enemies. You have to take away their crews too. Han Solo vs. Spock would be a great under card.
Hand to hand combat goes to Kirk. He grew up fighting in bars. Skywalker always fell back on the force. Luke has the light saber which Kirk would have no answer to when Skywalker used the force. But again we're taking the force away, so Luke would be left with a sword. And since Kirk is good with a gun, I'll take Kirk.
Mentally Kirk is a rock. He doesn't fear anything. That's what makes him so tough to beat. Luke might have the intelligence to use that against him too. Kirk once got inside Spock's head and forced him to resign as captain of the enterprise, only later to name him 1st officer. So Kirk can beat you mentally. And we all know that Luke is a head case. Luke's best chance here would be to stay controlled and bait Kirk into a overzealous action, putting Luke in a favorable position. But since I've never seen Kirk put in jeopardy, I can't assume Luke would get him there. So Kirk has the upper hand here too.
Weighing all of the options I do not see a way that Luke Skywalker could possibly take down James Kirk. I have to rule that Captain James T. Kirk is the winner.