Assuming we take both captains from their respective series and not in the EU or whatever Star Trek calls it, then Picard is obviously the better captain.
Kirk may have been cooler, but he was always fighting aliens with his bare hands or boning them with his bare.. well whatever.
That, although awesome, is not a good captain.
Picard on the other hand was always looking out for his crew and understood that sometimes you had to send your number one instead if the situation was too dangerous.
And he still did a ton of amazing things, without constantly endangering the Prime Directive (and his crew) as Kirk was wont to do.

Picard. Why? This: http://www.youtube.com/watch?v=xEZcslNOEBg

Darth Vader, obviously.

Captain Jean Luc Picard of the USS Enterprise!

http://www.youtube.com/watch?v=X6oUz1v17Uo

Definitely Picard!

Picard. Done.

This, the fact the womanizing was often with green women also makes him a visionary, WOW hadn't even been invented yet!

I say Kirk! Fortune rewards the bold.

Clearly a trick question. The real answer is Benjamin Sisko.

Fortunately, I have already purchased the 4th Humble Bundle, so I have no interest in being entered into this drawing. This means that I can provide the actual, correct answer for "who is the better captain?": Benjamin Sisko. He was a leader who got things done, and fought a war for his civilization. While Picard was off running around with luddite immortal hippies and whining about how he wanted to get back to exploring, Sisko was rushing to the front lines of the Dominion War and saving DEMOCRACY and FREEDOM. And while he did keep his shirt on, he managed to wrestle a reptilian man named Dukat until throwing him into a perpetual fire to burn for all eternity.

Also he was Bajoran Jesus.

Kirk , of course.

I am participating, because I don't have money to buy Humble Bundle passing the average price. I know, it sucks to not have 6 dollars for games. And I really would like to play Cave Story+. This is why Im participating.

This. In both accounts, ie I already have the bundle and Sisko was the best captain.
Kirk of course is the true and the only correct Star Trek captain, but that's another matter.

Captain James T. Kirk
he was always trying to prove humans are a decent species by having sexual relations with any female of any species. Nothing says equality like that.

Dunno, Kirk I guess as he came first. I'm not a trekkie, so I really have no bearing in this.

I'm not much a trekkie so i say Captain Kirk