I mean, you'd think that with how compassionate he's supposed to be, and with all his resources, at some point or another we would have seen him donating to an orphanage or building one or something similar. But I can't recall ever having seen him do so.
In the Nolanverse:
In The Dark Knight Rises, Bruce Wayne is identified as the primary sponsor of Gotham's Saint Swithin's Orphanage. Robin John Blake grew up there. In addition to footing the bills for the orphanage, Bruce visited it at least once, presumably to show his personal support for the children who lived there.
Blake: But we've met before. It was a long time ago. I was a kid. Uh, St. Swithin's. It used to be funded by the Wayne Foundation. It's an orphanage. My mom died when I was small. It was a car accident, I don't really remember it. But my dad got shot a couple years later over a gambling debt... and I remember that one just fine. Not a lot of people know what it feels like, do they? To be angry... in your bones. I mean, they understand. Foster parents. Everybody understands... for a while. And then they want the angry little kid to do something he knows he can't do. Move on. So after a while, they stop understanding. They send the angry kid to a boys' home. I figured it out too late. You gotta learn to hide the anger. Practice smiling in a mirror. It's like putting on a mask.
So you showed up this one day, in a cool car. Pretty girl on your arm. We were so excited. Bruce Wayne, billionaire orphan. I mean, we used to make up stories about you, man. Legends. And, you know, with the other kids, that's all it was, just stories, but.... Right when I saw you, I knew who you really were. I'd seen that look on your face before. It's the same one I taught myself. I don't know why you took the fall for Dent's murder... ...but I'm still a believer in the Batman... even if you're not.
Bruce Wayne: Why did you say that your boys' home used to be funded by the Wayne Foundation?
Blake: Because the money stopped. Might be time to get some fresh air. Start paying attention to the details.
After Batman's (fake) death at the end of the movie, we learn that Bruce left half of the rebuilt Wayne Manor (Alfred Pennyworth got the other half), and much of his fortune, to Saint Swithin's, and we can probably assume that the kids will be raised in the mansion.
In the Comic Books:
In the comics, he has taken it upon himself to raise no fewer than 3 orphans. These three were Dick Grayson (part of a circus performer family, his parents were killed while performing), Jason Todd (who has two origin stories - in the first, his story is identical to Grayson's - family of circus acrobats, parents killed mid-performance; in the second, he is an orphaned street urchin who Batman caught in the act of stealing the wheels from the Batmobile), and Tim Drake (whose parents were poisoned - his mother was killed outright; his father survived for a time, but was later murdered by Captain Boomerang). We know these characters better as Robins 1 through 3.
He is also Gotham's leading philanthropist, and among the many charities he oversees through the Wayne Foundation are a number of orphanages.
Through the Wayne Foundation and the organizations underneath, Wayne not only addresses social problems encouraging crime and assists victims in a way that his Batman persona cannot, but the arrangement also provides a large network of connections in the world of charities. He finds out about the newest trends and newest arts, and at the same time maintains connections to the streets through the soup kitchens and social services groups, which augments his crime fighting efforts.
The Martha Wayne Foundation is a patron foundation and supporter of arts, families, education and tolerance. The foundation supports and helps to run a number of orphanages and free schools, and provides teachers for those who have learning difficulties. Artists can apply for grants from the foundation to help support them in furthering the arts. The foundation sponsors companies like Family Finders Inc. in Gotham. Family Finders is an organization directed at finding lost people and uniting families. The foundation sponsors and runs dozens of soup kitchens in the city.
- Batman Wikia
In the 1960's Television Series:
Again, Bruce Wayne has adopted an orphan, most frequently referred to as "his ward" Dick Grayson, who serves as Batman's trusty sidekick Robin.