This is a tough one. I want to go with both but I think Cap would be the winner in the end. Both are superior fighters in the Marvel and DC universes, but I give this one to Captain America. While Batman has the greatest thinking mind, in an all out battle no one tops Cap. He was trained to be a super soldier who could overcome anyone in battle. Cap beat the Nazi's and Adolf Hitler for crying out loud!
As good as Batman is, he really isn't as good a fighter as Captain America. All his extra gadgets wouldn't do much either, if Cap can avoid bullets, he can avoid a simple batarang. DC themselves admitted to this when the first DC/Marvel crossover took place. If you think of this in terms of if this were a real fight, and not on who you like better, then you'd see Cap winning. Batman might win the vote on being more popular, but Captain America could beat him in a fight.
They've both got their smarts and have proven themselves in their strength and abilities but Cap is a step beyond Batman with his physical strength and determination. The man survived being frozen for goodness sake! It basically comes down to this, Captain America is an enhanced human fighting machine due to the super soldier serum. Batman is merely a highly trained schizophrenic. Captain America is the superior hero.
Sorry Bats, love ya, but Cap would win the fight by ramming that sheild up the bat's ____ and twisting it. Patriotism at it's best!