Manhattan, especially the midtown, is probably the dirtiest part of New York City. Better subway stations in Manhattan? You must be kidding, right?
New York had its share of democratic mayors. New York had a black mayor. I doubt they all were trying to keep non-white people at bay.
Racial issues do exist and they do affect many parts of American lives, but they should not be used to explain all the problems of American society.
I've overly simplified why American cities are the way they are. It's not just racial issues, but general lack of public funding, political reasons, etc. Having said that, places like London and Tokyo don't have the same overt racial issues as NYC. I often see homeless people and panhandlers in the subway who are down on their luck, often with psychological problems. They are frequently non-white and it's sad that the city isn't taking care of them. I don't see that in large international cities in other developed countries to the same extent. In London, for example, I rarely see non-white people in such dire circumstances in public areas compared to NYC.
My general point is that cities like London, Sydney and Tokyo are cleaner, nicer and have better public services. Personally, I would feel more comfortable raising a family in those cities rather than a U.S. city. In the U.S. with a family I think it's better to raise a family in the suburbs, but that's my personal preference.