I'm going to start this by saying that Batman is a notorious loner that hangs out with A LOT of people. He has had a multitude of sidekicks, been on many teams, and started many of his own, but shockingly enough his direct family has been lacking in diversity.
Out of his direct family we have all the Robins which were white carbon copies of each other (features at least). They tried to make Jason Todd blond but retconed it. The only person out of the Robins that has any form of diversity is Damian Wayne who is biracial. He is half white and half whatever Talia is. But unfortunately DC hasn't realized the potential of that character to play on it, there aren't many multi-ethnic characters out there since people like to make "racial pairs" like Storm and Black Panther, or Black Canary and Green Arrow (and you thought it was just a coincidence they were both blonde, out of the fewer blonde characters out there). Damian Wayne has been treated and drawn like a white person so for all intensive purposes he is white.
Then you have the Batgirls, Barbara who is a red head (which I wish was an example of diversity but red headed females are REALLY common in comics and cartoons), Stephanie who is the only Blonde character in the bat family so she gets points for that (I don't know much of the character and I don't know if she has dealt with any stereotyping due to her hair but I doubt it). And then you have Cassie! The first real step to any diversity, she is Half-Chinese and Half-White. She is almost Damian 1.0, but we actually were aware of her heritage. Unfortunately DC's editorial staff didn't like her tried to ruin her and now got rid of her. They also did the same with Stephanie brown too...
Outside of his family Batman's main team has always been the Justice League, and if I ask you to name who are the members of the Justice League here are the names that might come up: Superman, Wonder Woman, Green Lantern, The Flash, Aquaman, Martian Manhunter, Green Arrow, Black Canary, Hawkgirl (the TV show had a lot of influence), and if you have been reading DCnU Cyborg. My point being that you either have to be white to really be on the team for a long time. Multiple attempts have been made to diversify the team the two most famous is for JL cartoon by using John Stewart and now with Cyborg but overall the trend is the character needs to be white (btw Martian Manhunter's alter ego is a white guy). It actually gets even worse with the Justice League because its made up of elitists. You have your billionaires (Batman and Green Arrow), your Royalty (Wonder Woman, and Aquaman), and your superpowers and I mean they are the MOST powerful just by power set alone (Superman, Martian Manhunter, Green Lantern, and the Flash) who also happen to be connected to the military/police in some way (except for Superman).
So where have people tried to enter diversity into Batman's world? By his extended teams.
The first real major attempt was the Outsiders which had Black Lightning (he's black), Katana (can you guess that she's japanese), Geo-Force (ok this one isn't obvious, eastern european), Metamorpho (he's a rainbow of colors!), and Halo (who is this naive girl, can you guess what her hair color is). This is a much nobler attempt to diversify Batman's world but the problem is why not introduce someone directly into his family. They have to be extended and from another country. This is repeated in JLI and Batman Inc. All attempts to give us a black Batman, or a brazilian team mate. But thats the thing, they are all from other countries, they don't hail from gotham (other than Black Lightning) but even so they don't work with him in Gotham. They are separate worlds essentially. He has his white gotham, and diverse groups outside of the country.
Has the Bat Family made any headway into diversity. I would argue yes, and definitely in the gender and sexuality front. Batman's world has been very good at including strong independent women. At a time there were even more females (Oracle, Huntress, and Batgirl) than males (Nightwing and Robin) that were part of the family. Oracle becoming a major player in all DC, and I would say is the only Bat character that didn't remain in the shadows of Batman. Thats pretty impressive and empowering to women. And on top of that she has been the number one Superhero on a wheel chair for the past 2 decades (and totally outshining Professor X). And since I am at it with people that have disabilities Batman also had a genius mute named Harold working with him. You also have a lot of growth in the Lesbian front with Batwoman (who was ironically introduced to fight claims of Batman's homosexuality, brilliant move DC!), Maggie Sawyer, and Renee Montoya. And best part is I don't feel that their sexuality rules their character so I definitely applaud for that. Though you definitely see how their sexuality plays a role and affects their life.
There is a lot of growth even in the sphere of sexuality though. Gay and bisexual characters are practically non existant, so here's to hoping Harper Row and her brother develop and grow.
Why is this a problem?
1. I am actually going to make a list for this cause there are a multitude of reasons. The first is simple representation and subliminal messaging. Without decent representation of different groups then you get subliminal messaging of the represented groups superiority. A very real example is Superman was first introduced to represent white america. Many of his inicial villains were foreign, and what did he fight for again, Truth, Justice, and The American Way. So there are problems when you constantly present rich white people as the saviors. It might sound ridiculous but the affects of subliminal messaging is powerful, when all you read of black people in the comics are is that they are poor people that need saving, or thugs that rape and beat innocent people (never the crime boss though). It sends this subliminal message that black people are inferior and need white people to save and protect them, that or they are dangerous and avoid them. Plus it ends the perpetuation that Americans are white and anyone else is foreign. Last time I checked everyone tracks their heritage from another country, except Native Americans who are practically extinct (if a group of people can be extinct).
2. Here is an economic reason, if you create characters that have different ethnicities, then people of those ethnicities might start being interested in reading that character. I mean Wonder Woman was an icon for women, and attracted many female readers. Logic doesn't fail that if you create and Arab superhero that more Arabs would start reading it (especially cause the hype around Arab superheroes is usually so dramatic).
3. Batman deals with characters with many psychological issues, heroes and villains. There is a hugely untapped area of stories that are all around diversity and the psychological issues that surround it. A variety of identity issues come up from growing up as a Black person in a white society, constantly having to straighten your hair, dress like a white person, talk like a white person, simply to fit in. I won't go into a rant about it but drug laws highly target African Americans. Latino and Asian-Americans are constantly being used for cheap labor. The issue of being bi racial where people perceive you to be one thing when really you are both. Transgender issues where people are simply just uncomfortable being around you. The double sided discrimination Bi-Sexuals get from both the Homo and Hetero sexual communities. The fact that no matter how many generations your family has been in America Asian's will always be viewed as Asians and not Americans. And the list goes on, and all of it is very juicy untapped character stories.
Now I am not saying diversity for diversities sake is a good thing, in fact that can lead to one note boring characters. But entering diverse characters into the Bat Family can lead to very intriguing story lines. I mean one of my favorite stories that I read was actually how this Arab-American who has the powers of storm is in a plane that is about to fall. He attempts to save the plane but at the same time as Power Girl and Batman, and everything thinks he was the one bringing down the plane. Dealing with his fears, and and what happened to him was REALLY interesting and REALLY real. And it was different from everything else I have read regarding Batman. We've seen the mugging, we've seen the bank robberies, we've seen the murders (all blue collar crimes btw...) but we don't ever see that! And thats what I want!