The area around Journal Sq is not to bad. McGinley Square (roughly 10 min walk to journal sq) is a good option as well. Its not as safe as downtown but its not a warzone either. Downtown is the place where most people move to. However, you may have a tough time finding a place in downtown and Hoboken with that budget. You may be able to find a small 1 BR or studio apartment. Washington St area is downtown and is safe. You just have to deal with a lot of drunk people walking around at night. If you have a car, parking is a major pain. As for diversity, the entire city is a melting pot so you should be ok.