Batman and Captain America have rarely crossed paths in the comics. However, one Marvel and DC Comics Elseworlds crossover paired up the Caped Crusader and Cap in a story that saw the unlikely duo in Gotham to take down both the Joker and the Red Skull.
In 1996, Marvel and DC Comics released the Elseworlds story Batman & Captain America, penned and drawn by John Byrne with colors by Patricia Mulvihill, that saw the Golden Age versions of Captain America and Bucky Barnes end up in Gotham for a top-secret mission surrounding the Gotham Project. After Batman saved Captain America from freefalling off the side of a plane, the two fight soldiers who've been experimented on with poison patches by the Joker. As Batman investigates, Steve Rogers is assigned as Bruce Wayne's bodyguard as he's initially thought to be connected to the Gotham Project.
Rogers sits outside Wayne Manor for days with no attacks on Bruce. However, when Steve overhears Bruce telling Dick Grayson that he needs to "find a way to shed private rogers and keep my appointment with the Joker!" Steve starts to believe Bruce might be a bad guy after all. He follows Bruce to the Wayne Foundation Building and confronts him. The two exchange blows before they both realize they aren't fighting just a normal person but fighting an opponent with incredible skill. In a hilarious moment, the two stop fighting and shake hands as Steve realizes Bruce is Batman and Batman realizes Steve is Captain America.
Meanwhile, it's revealed the Joker is working with Red Skull (for a sumptuous reward). After Batman and Robin are taken hostage while investigating one of the Joker's supposed hideouts, Captain America realizes that the Red Skull is involved. The Red Skull loads a giant bomb on a plane and is greeted by the Joker. However, when Red Skull reveals his Nazi origins, the Joker takes a stand against him: "I may be a criminal lunatic, but I'm an American criminal lunatic." The villains try to kill each other but are immune to each other's similar venom concoctions. Cap and Bucky free Batman and Robin and chase the bomb-wielding plane in the Batplane.
Captain America, Bucky, Batman, and Robin get onto the plane, but it appears they are too late, as Red Skull is close to dropping the bomb and killing and infecting thousands. Joker, in an act of patriotism, launches the bomb early with both him and Red Skull falling into the water with it - exploding, presumably killing both of the villains. The book ends with Batman suggesting that Red Skull and Joker have finally been defeated. Cap gives him a look and says "you don't really believe that, do you?" Batman responds "No, Captain. I'm afraid I don't..."
Batman & Captain America is one of the better Elseworlds titles as the two heroes (and their sidekicks) bounce off of each other perfectly. Both heroes have similar moral compasses and mindsets to being a hero. Meanwhile, the power dynamic between the Joker and Red Skull is chaotic and ultimately leads to their demise - which is likely the only outcome for the two power-hungry psychopaths. In terms of DC/Marvel crossovers, it doesn't get much better.