I went to a very diverse High School in New Jersey (Union County). Although I knew many people of different races, and I never considered myself a racist, everyone seemed to stick to their own race, as far as social circles go.
I never saw it as an issue, because if I saw a guy from High School I knew, be he White, African American, Asian, Hispanic, etc. in a bar, I would sit down and have a drink with them, and catch up on old times.
I feel that people tend to stick to their own kind, not because of racism, but simply because of cultural similarities; ethnic or just social cultural differences tend to have people group together.
Furthermore, many people I knew from High School were racists, not because of their parent's upbringing (many of their parents had friends of many races, and I never saw any behavior to denote otherwise), but simply because when one race dominates the population in an organization, such as a school, people become suspicious of them, and begin to feel "outnumbered" and threatened.
I don't believe this causes racism in ALL people, or even MOST people. I believe this simply adds a few bigots who like to be in control, and have an overall feeling of superiority.
As far as raising children goes, don't go out of your way to put a child into a diverse school. If the school across the street that is mostly white, also happens to be one of the best schools in town, send them to that school. On the other hand, if a school across town is just as good as the one across the street, and is much more diverse, you may want to consider sending your child there.
Being so fearful of raising a racist, and preventing your child from being "capable" of bigotry, may cause a level of rebellion in your child, who wants nothing more than to "annoy you". Eventually these feigned beliefs will become common place in their minds, and could also lead to them being permanently racist.
My verdict:
Allow your children to have equal access to multiple cultures, either in school, extra curricular (churches, sports teams, private organizations), or at home with entertainment. Preaching to your child will just make them want to act differently than what you want them to. Don't you remember saying "I'm NEVER going to do xyz to my kid, like my mom/dad did to me!" The same rule applies here.
In addition, children cannot be forced one way or another based on observations alone. Children mimic their parents only for a few years, and once they hit their teens, they will attempt to be "different" from everyone else. If you are neutral to other races, not preferring, or deterring a particular culture/race, I believe your children will simply make up their own mind on the subject.
I'm not an expert, and I've certainly blabbed a lot, but I think this is very true among most children.
