I think I would start out trying to reason with her. Try to explain, calmly, that blacks and Hispanics are people too, and all people have certain rights. She may have never actually heard that side of the story! Maybe show her an example, a black or Hispanic person who works hard, who does something interesting or important or heroic, and then say 'Don't you think he/she deserves xxxx just as much as anyone else who does that?' Tell her she hurts your feelings when she says stuff like that, and does she really mean to do that? Tell her when she's being rude. Kids often just don't know when they're being rude! And when she's rude to -you-, well -you- are the grown-up here and you won't put up with it.
If she likes you, if she respects you at all, you might have an effect on her. Through both talk and your own personal example of how you treat people, if she sees that. You'd be surprised how much influence a babysitter can have on a little kid.
If that doesn't work, i think I'd have a talk with her parents. They might have taught her that without really meaning to. If -they're- racist, then I wouldn't work for them..