Maria, the suburbs around NYC are all really pretty diverse and generally speaking, tolerant. I have a lot of family (white) in Oceanside, and some of them have some subconscious racism ... but everyone is appreciative of the diversity in race, religion and personality that you'll find in and around Oceanside. There are nearby communities with a much higher percentage of black, Latino, etc families (I think Malvern and Freeport spring to mind as vibrant communities with a wider array of skin tones than Oceanside) but Oceanside does have fabulous schools!
I also find that kids on Long Island grow up meeting other kids from all over the island - there are so many family oriented things to do, you will find yourself leaving Oceanside for birthday parties, social events, etc...so your kids won't grow up only seeing just white face after white face.
On a personal level, I would say you don't have to fear any outright racism - certainly not any sort of violence at all! In fact, I think you will find a city filled with very open minded and tolerant people. But of course, the subconscious racism kids encounter can be just as damaging, but I think you will find that absolutely anywhere in our country if you're looking for it.