I'm happy to say that my gut feeling is the opposite Paul.
I grew up and started school shortly after desegregation. I started the first grade in 1971. I was around black kids throughout my grade school career, unlike my parents. We had more contact than any generation before us. This has extended through my life. I've had female, black and black female bosses. 20 to 30 years ago I think that would have been highly unusual. It's not uncommon today. Hell, we have a black female Secretary of State (who is lousy, which has nothing to do with her race or gender) who replaced a black male, who replaced a white female.
The culture has changed. It was common for me to hear overtly racist comments from adults when I was growing up. I don't think my children ever have. The mass media depicts more diversity in a less self-conscious manner.
I believe everybody has prejudices, some of them racial. I believe that my generation is more aware of mine and do more to fight them than my parents' generation did. I believe my kids will do better than I did.
I'm not saying things are perfect, or even close to where they should be. But I do believe things are getting better, gradually.