One of the things I like about Batman is how his family is made up of people from all over the class map. Yes, Jason was the most badly treated Robin editorially--and that is related to his class in that his second incarnation was specifically made to be a street kid. But that also went along with him being a rebellious kid mistrustful of authority (as opposed to an Oliver Twist type who looked up to Bruce for saving him, for instance). His real problems as a character--I mean, the way that he is now, being an anti-hero or villain--come from his getting killed by the Joker. The character himself has now grasped onto class differences as part of his whole narrative of Bruce never caring about him--and I buy that and it's interesting to me--but a lot of it comes down to Jason's issues.
I don't know--I mean, of the Robins the two most successful ones come from different class backgrounds, whether or not Dick's also an ethnic Romany. Barbara's the daughter of a cop. Alfred's a butler.
Bruce does come out of what I think was a common trope in the 30s/40s of the millionaire who fought crime, and his money explains a lot of what he tries to do. I like some of the subtle patterns that have come out as people write the different characters responding slightly differently to Bruce's super wealth. But it seems especially wrong to accuse Bruce of being a rich guy who beats up poor people when, even leaving aside the fact that all his villains aren't poor or created by a life of poverty, and that he often protects poor people both as Bruce and as Batman, the guy's taken kids from different classes and made them his sons. It wasn't until Damian that there was any suggestion they things Bruce was born with mattered so much. Tim's being from the upper class was obviously a conscious choice and part of his character, but it never put him above Dick the carny kid.