I live in New York City, so I can talk about Columbia and NYU.
NYU is located in the West Village, which is a relatively safe area. It's mostly residential and largely populated with NYU students, so it is safer than some other areas of the city. It's also a huge tourist destination, so it is usually pretty crowded and there is a strong police presence in the area. Plus, NYU doesn't have a true campus, so university police are stationed throughout the area.
Columbia is way uptown, and while the area surrounding Columbia is okay, be careful stepping too far away from campus, especially if you are not familiar with the city. However, the area up there is lovely and as long as you watch where your going and pay attention to your surroundings you should be fine.
The key thing to remember about going to school in any city is that it is as safe as you make it. Always be aware of your surroundings and your valuables. Use bags that close at the top with a zipper or buckled flap and keep your eyes and ears open.