Parts of Harlem are pretty seedy; and I can't say I'm too familiar with the neighborhood to tell you about it. But she'll stand out like a sore thumb there, but a young woman there, alone will stand out in any city. You want my advice? Tell her to stay in one of the cheap places in New Jersey or outside of the city. Everything in New York is going to be very, very expensive and/or seedy as hell. There's a YMCA in Greenpont (Brooklyn) that I found on hostels.com. It's a pretty safe neighborhood.
lol, a hostel in Greenpoint or any hotel near any airport is on no way safer than her former situation in Harlem. and did I read correctly, where did it say she was traveling alone? the OP mentioned her daughter and that "they" are traveling together. they will be fine.
The most important thing to tell her is to not try to go to Rome, Florence and Venice for 2 nights each. Scrap Venice and spend those days in Rome. Two nights in Rome isn't even close to enough. I cannot stress this enough. She is trying to squeeze too much in, and will end up lessening the overall experience. I will say it again...SCRAP VENICE. Venice is a tourist nightmare anyway, although it's worth seeing if you have the time. And don't sweat the safety issue. Just tell her to use a leg wallet and she'll be fine. http://www.amazon.com/Eagle-Creek-T...=1398362256&sr=8-1&keywords=travel+leg+wallet