I'm not sure what it's like now, but when I was considering going there, I was told they only take around 10 "Tech track" people each year. Tech track is what stage management falls under in the program- there is no separate stage management major, only an emphasis in the large design/tech field.
I have several friends who did eventually go to NYU, including one who is in the tech track. She is a stage manager, but also very interested in costume design, so it works well for her. If you aren't as interested in the other design/tech areas, I would discourage NYU. But it's a great school if you are, and can afford it.