Easily Kirk. Unless Sisko prays really hard and gets the wormhole aliens to magic Kirk into nothingness.
Classic Kirk in his prime had the whole Kirk Fu thing going...few in the era formerly called modern Trek could compete with that. Sisko would be giving nuKirk atomic wedgies.
Surely they'd be on the same side? I think if it did happen though, Kirk would win if it was Sisko making a guest appearance on TOS and Sisko would win if it was the other way around. You've got to win on your own show!
Kirk beats early Sisko. Bald Sisko with facial hair will take all of you out, then whip Kirk's butt in spite of the prophets telling him not to.
I think the fight would turn out like the end of The Quiet Man. They'd kick each other's butts for a long time, neither would be willing to stay down, and they'd end up having a drink together.
To answer the question properly, we needed to know why they fight. Is it a reason they would die for, or is it something where one of them (or both) back up and say "Ok, let's stop."
Shat is only like 5'10" (advertised, I think he may be shorter than that in reality) and in his prime, was about 175 LB. Brooks is 6' 1" and in his prime was about 200 LB. Based on that, I would say Brooks has better reach and is stronger. In real world, Brooks would take Shat, but since we are talking characters... Kirk has his Kirk Fu and as soon as the shirt rips, it's all over. Sisko may have decked Q, but Kirk bitch slapped Trelane (who could have made Kirk just as dead just as fast as Q could have with Sisko), Karate chopped Gary Mitchell (same as Trelane), laughed in the face of Apollo, pnwed Khan in a fist fight, laid some smackdown on Captain Ronald Tracy, kicked lord Kruge off a cliff, and held his own with Spock, and a boulder-hurling Lizard man!
The setting would matter, too. Kirk in his prime in the TOS engineering room...no contest. Kirk won against a genetic superman with five times his strength by beating him senseless with a little plastic pipe.