NYC at the end of December..?

Due to my workplace's new "use it or lose it" policy on time off, I have several days off that I need to take this month.

I'm thinking about going up to New York City around Christmas, though I plan to be home for the day itself.

If you live in or are familiar with the city, could you give me some advice? I'm mainly interested in going to museums and performance/art places, instead of shopping (though I'm might try to find the millinery supply place I've heard about) or big tourist attractions.

I'm also on a budget, so I'd appreciate suggestions of the safer places to stay that don't cost too much and are close to decent/uncomplicated-for-the-visitor mass transit options. I'll probably take the train or an intercity bus up if I do go.