NYU area is very safe. E. 23rd street area, used to be a bit rough in the 70's, now it's solid. Got Murray Hill to the North (a lot of young bankers live there), and Union Square to the South.
Columbia is a slightly more "sketch." It's in Washington Heights/Morningside, but that area is gentrifying rapidly as Columbia is buying up property left and right up there. Just exercise discretion and you'll be fine.