As for safety, New York, Philadelphia and Washington DC are the same as any big city in most countries. There are certain neighborhoods where it would be unwise to go walking alone late at night. This would be the same in London, Liverpool or Paris. Those unsafe neighborhoods are typically not the tourist areas of the city, so you are unlikely to venture into them anyway.
The Youth Hostel in New York City is on the Upper West Side. This tends to be an ethnically diverse area similar to the way the West End of London used to be years ago. (I don't know if that is still true today.) I don't think it is any more or less safe than any other part of Manhattan. The Hostel is only a few blocks from Broadway with its open-all-night businesses and its round-the-clock police presence. Some of the side streets in that neighborhood might be a bit dark and intimidating at night, but you can avoid them.
Washington DC and New York City both offer many attractions including cultural, entertainment, historic, shopping and dining. Philadelphia offers less. It has the historic downtown area and some cultural and entertainment attractions but not on the scale of the other two cities. Baltimore offers some nice attractions near the Inner Harbor area of the city, and you can easily commute there in less than an hour from DC.