Meh. I'm ok with it. I'm a member of the local YMCA, and as soon as I can afford to, they're close to the top of my list of local organizations I would like to support. I have no problem with a name change that makes women or non-Christians feel more welcome.
I can say that the first time I went to a YMCA--to participate in a judo group that met there--I felt very uncomfortable. I really had no idea if Jews were welcome, or women, for that matter. Of course, I practically grew up in a cave and was wary of all kinds of odd things, so I don't know how normal my feelings were.
But a close friend of mine--also a former Orthodox Jew--seriously needs some support/encouragement/opportunities for exercise, and every time I push her to join the Y, she counters that she's meaning to join the JCC (Jewish Community Center), but there are money problems. The Y is cheaper and has better facilities, and would be much healthier for her, without all the Orthodox people there who disapprove of her, and yet she is reluctant to join. I think that she certainly fits into the scope of the Y's mission, regardless of ethnic identity, and a name change that makes her feel less unwelcome seems like a good plan.