Tell her you guys have "accepted" it and that you really want the whole experience to be good so that the trip will have lasting positive memories. Sell her a bit on New York and how fun staying in Manhattan could be. As a gift, offer her a hotel room if eh changes for a later flight to enjoy New York for a day. And the tips about a couple key things sounds smart, too.
Just tell her that you'll pay for her hotel in New York, the safety should outweigh the 20 bucks saved. I'm sure she would agree.
huh huh huh... sounds like your kid has her s*** together and is more responsible than most people her age. there are plenty of young dumbasses who travel abroad and make it back...she will be absolutely fine in italy and have the time of her life...sounds like she earned it! this!
I think the OP's main concern is his daughter staying at the Y in Harlem. I think most that read his post and understood that, agree with him as well that staying at the YMCA in Harlem isn't the best idea for her. Whilst we can all agree she has done well in paying for a trip to Italy for herself, that YMCA in Harlem thing is an issue. Even is she doesn't want your permission or help, she shouldn't quibble if all you want to contribute is for her to stay at a better hotel(or hostel) in New York City. She can do and go wherever the heck she wants but your daughter shouldn't be stubborn about the hotel or hostel in New York, especially if you're offering to pay for just that.
Some in different cities allow the homeless to stay there overnight. I'm guessing others allow visitors to pay to stay there overnight. Either way, there are hostels in such cities that are better options if one can't afford an actual hotel.
Give her the gift of the hotel room. Tell her it's always an option. Perhaps they change their mind last second when they see that staying at a YMCA is not all that great.
Instead of just objecting to her staying in Harlem, try to understand why she wants to stay in Harlem. I for one must admit I don't understand what's attracting her to it, and this is coming from someone who lived in the vicinity for a few years around her age or a bit older. It's actually a pain to get to the fun places from up there. Understand what's her fascination with it, then come back here and ask for suggestions of better things to do and not better places to stay, because you're never going to convince her through arguing the latter.
Harlem YMCA is a hostel. she probably found it as one of the first searches for NYC hostels. she should try the Jazz Hostel along Central Park West and 106th, Duke Ellington Blvd. it would put her in a place where, simply, there is more to do in one night. that being said, people hear the word Harlem and think it is some sort of war zone. the jazz hostel is practically Harlem anyway but closer to museums, better subway access, and like I said, it's on Central Park.
Were is she going? I agree that there is nothing to worry about in Italy. This is the best advice in this thread so far. Try to figure out why she wants to stay at the YMCA. And discuss with het what options she has (maybe what she likes about the place is also possible in a safer hotel).
Honestly the YMCA thing would scare the hell out of me too, but the website doesn't look too sketch at least - http://www.ymcanyc.org/association/guest-rooms/harlemrooms
My sister has traveled all over the world (including Africa and Asia) solo or with other girl friends (some of whom she met on the road) and had no issues with crime. She will be fine in Italy and the most dangerous thing she'll find there are Italian boys (which she is probably COUNTING on). As long as she has common sense and uses good judgement she'll be okay.