I know your instant answer to that question is, “No! Of course not!” but consider the possibility with me for just a few minutes. We know that Batman is willing to fight by the side of any hero when push comes to shove. He is a prominent member of the Justice League which is filled with supernatural heroes, yet when it comes to Bruce's inner circle, there is a mysterious lack of superpowered beings. Batman has adopted five children, yet none of them possess any powers. Almost all Bruce's closest allies are powerless. Bruce zealously guards Gotham from unapproved vigilantes, and almost every hero he allows to operate in Gotham is unpowered. Even when he offered the support of Batman to the whole world through Batman Incorporated, Bruce basically put a sign on his club house door that said, “No Metahumans Allowed.”
With all this in mind, I think it is fair to consider the question. I mean, aren't there any young, vengeful, orphans with superpowers that Batman could lead into a life of violence and vigilantism?
Again, the vast majority of Batman's allies are humans, but with a rolodex as full as Batman's, we can find all kinds of characters. All of the Robins and Batgirls are unpowered, but Catwoman is suspected to have an empathic affinity for cats. Azrael (Valley) had superhuman strength due to his line of strong genetic forefathers. The Birds of Prey operates under tacit approval of Batman, yet it is led by Black Canary who possesses a skull shattering canary cry. Even though these are exceptions to the no power rule, these are still relatively minor powers considering the amped up power houses that live in the DC universe.
Speaking of amped up powerhouses, Superman causes a significant chink in the theory that Batman is bigoted. After all, Batman considers a mighty alien to be his best friend, and Bruce often teams up with Clark even outside of their work with the Justice League. However, Batman is famed for his suspicious nature, and you could easily conclude that Bruce is only close to Superman so that he can be there to take him out should the need arise. Sure, Batman calls Clark friend, but Bruce is hardly above lying.
Batman Incorporated also has some exceptions. There is an entire unit of Batman Incorporated called the Dead Heroes Club that includes heroes with super powers. Also, Squire has the ability to interpret information by touch. Still, it remains true that not a single superpowered being is allowed to be “Batman.” Batman Incorporated's limited inclusion of metahumans might seem to disprove the idea of Bruce's bigotry at first, but if you apply Bruce's decisions to the real world, it makes them appear even worse. Can you imagine if the military created a separate branch for black soldiers or if women were prohibited from attaining a certain rank?
I've been playing Devil's advocate, but let's get real for a moment. We all know why Batman does not have a bunch of superpowered allies, and it has nothing to do with bigotry. Batman is light on metahuman allies because it would ruin the tone of Batman. There are very few superheroes without superpowers, and none are as well known or well respected as the Dark Knight. It is the gritty determination of an unpowered man to focus him mind and body to overcome any obstacle that makes Bruce an admirable character. It is the knowledge that Batman cannot shrug off bullets and the lowliest thug could potentially cripple or kill him that adds tension to his battles for justice. If Batman had an ally that could simply throw up an impenetrable, psychic shield, it would fundamentally alter the nature of Batman stories, and nobody wants to see that.
However, that is an outside the universe answer to the question. What is the inside universe answer?
The In Universe Perspective
As far as I know, this subject has never really been covered in comics, but here are some theories that could explain Bruce's bias.
1. Bruce doesn't trust metahumans. Perhaps Bruce's feelings about metahumans are much the same as his feeling on guns. They can be useful, but they can also be extremely dangerous and unpredictable. It is better not to add an extra layer of risk to his operations. He knows the capabilities of humans, but metahumans are a wild card that cannot always be properly predicted.
2. The unpowered need help to fight crime. If you can walk through a hail of gunfire and punch a hole in a mountain, you probably do not need a billion dollars of Wayne tech to give you an edge. Powered heroes can go out and start making a positive change in the world just by virtue of their natural abilities, but human heroes need equipment and training, two things that Batman can provide.
3. There are better places for metahumans to learn to be a hero. Batman doesn't know what it feels like to be able to shoot beams out of his eyeballs. He has no idea what it is like to have to pull your punch because a fraction of your strength could take off a villain's head. Batman simply does not have the experience to help people like that, but there are others who do. The Justice League, throughout most of its existence, has reached out to new heroes and helped them out. There are tons of teams which take fledgling superheroes and give them more experience with the moderate safety net of allies. This was especially true during the time the JSA was in operation for the JSA were essentially superhero trainers. If you had powers, why would you look up to Batman to train you?
4. Batman is a symbol. Criminals are a cowardly and superstitious lot, so what makes you more scared if you are a street level criminal? A guy in red and blue who can punch through a mountain or a guy who can blend in with the night and take you out with a punch to the face. Superhumans generally handle superhuman threats. They don't worry about the common lowlife. Having a hero who is human reminds both good guys and bad guys alike that everybody can be Batman. Your neighbor across the street, the police officer on the corner, and the person reading the newspaper on the park bench could all be someone watching you and waiting for you to commit a crime just so they can punch your lights out. The metahumans are going to fly right over your head and punch Godzilla in the jaw, but Batman is probably watching you right now.
I don't think Batman is a bigot. At most, he is more suspicious of superhumans because of their greater power. However, this idea is largely unexplored in comics, and it could lead to some interesting plot lines if Batman had to deal with a new powered trainee or he was called to task on his apparent bias. I would not like to see Batman surrounded by metahumans, but as a brief story arch, it could be interesting.