Batman is always prepared for every eventually, and if given sufficient prep time, can defeat any opponent.The Doctor never plans anything, but is perhaps the greatest tactical improviser in the universe. (Doctor #7 is the exception to this rule, and he's probably the most dangerous and calculating Doctor as a result.) Advantage: The Doctor. There is simply no plan in the history of the universe that the Doctor couldn't figure out some way to foil, even if Batman had his whole lifetime to prepare. The Master has had many lifetimes.
Batman's opposite number is the Joker, the clown prince of crime, the human incarnation of chaos and meaningless cruelty. He possesses limitless ingenuity and cunning, and is personally responsible for thousands of deaths.The Doctor's opposite number is, like him, a nameless member of the ancient Gallifreyan race, with many lifetime's worth of advanced science and learning at his disposal. Unlike the Doctor, however, The Master has dedicated his many lifetimes to the pursuit of bending the entire universe to his diabolical will - or, barring conquest, he will be perfectly content merely to destroy all of creation.Advantage: The Doctor. The Joker's best moments were killing Batman's sidekick and paralyzing one of his most trusted allies. The Master's best moments were laying waste to the entire planet Earth with flying vivisection robots from the end of time; and, oh yeah, let's not forget he annihilated a significant portion of the universe in Logopolis.
Batman is a master of every martial art on the planet Earth. The Doctor only knows Venusian Aikido, and he probably forgot how to do that half-a-dozen regenerations ago. However, if pushed into a corner and forced to fight, he can learn any physical skill in as much time as it takes him to observe his opponent's most rudimentary moves. Advantage: The Doctor. Sure, there's no question that Batman knows more about fighting, but Batman would have to incapacitate the Doctor almost instantly for this to mean anything. Otherwise - as during the climactic swordfight in The Androids of Tara - the Doctor would be able to adapt to and counter Batman's best moves in less time than it would take Batman to size up his strange opponent.
Batman is the world's greatest detective, and additionally no slouch as a mechanic, inventor or computer expert.The Doctor is perhaps the most brilliant scientific mind in the universe. Advantage: The Doctor. He would also probably know better than to build a killer satellite using Brainiac technology that would eventually go rogue and try to conquer the planet.
Batman surrounds himself with an army of sidekicks and allies, overcoming his all-too human limitations through the power of teamwork and the support of his ad hoc extended family. The Doctor travels with companions as a way of alleviating the loneliness of constant travel and isolation, befriending creatures from across the universe as a means of distracting himself from the burdens and responsibilities of practical immortality. Advantage: The Doctor. Even the Doctor's lamest companion, Adric, still died trying to save the Earth; Batman's lamest sidekick died getting hit on the head with a crowbar a bunch of times by the Joker. And there's no doubt, Leela, K-9, Ace and Donna Noble are way cooler than Dick Grayson and Tim Drake.
Batman drives around in a state-of-the-art automobile. The Doctor travels across the universe in the universe's last functioning TARDIS, even if the navigation console and chameleon circuit never quite work like they should. Advantage: The Doctor. Come on.
Batman's alter ego is billionaire playboy Bruce Wayne. The Doctor doesn't need an alter ego, he just switches faces ever 50 or 75 years to keep things interesting (his brain gets a bit fried in the process, but that's OK, he didn't need to remember Peri anyway). Advantage: The Doctor. The Doctor has the additional advantage that every time an actor who played the Doctor appears in another movie or television role, the viewer can pretend it's secretly the Doctor in mufti as part of some elaborate ruse occurring off-camera. For example, All Creatures Great and Small is even more fun if you pretend that Tristan Farnon is really the Doctor, slumming in Darrowby as a means of foiling the Daleks' latest invasion. The closest you get with Batman is pretending that Batman is really Patrick Bateman during The Dark Knight.
Batman has an ongoing, on-again / off-again sexual relationship with Catwoman; as well as intermittent romances with the likes of Zatanna, Wonder Woman and Vicki Vale; as well as, in his guise of Bruce Wayne, a host of nameless starlets and debutantes. The Doctor . . . well, um, he's got River Moon, sort-of, perhaps, in the future. And if you read between the lines, maybe Romana. We know he had sex at least once in order to procreate, but probably not with a human woman. Advantage: Well, OK, this one goes to Batman.
But still, there can be no doubt, the winner is . . .
"I'm afraid, my dear boy, I simply have no time to be spent
gallivanting around with men dressed in rodent costumes."