I cannot help you with accommodations.

Crime on the NYC subways increased last year, but a recent beefing-up of police presence on the subways is said to have brought the numbers down somewhat.

From NYU to Columbia, the uptown #1 train from Christopher Street Station to 116th Street Station is about a 25-minute ride, and is almost certainly the best way to go.

You can view NYC subway and bus maps online at:

https://new.mta.info/maps

Free, full-size subway maps are available at any subway station customer-service booth (although sometimes they run out); free bus maps for the Borough of Manhattan should also be available at Manhattan subway station booths. Both subway and bus maps might also be available for free in the ground transportation area at your arriving NYC airport terminal. I recommend getting both the subway map and the Manhattan bus map.

Subway and bus fares can be paid with a Metrocard, sold in subway stations and by retailers (including some in the airport); fares can also be paid with contactless credit/debit cards and mobile-wallet devices. Up to four fares can be paid at one time with a Metrocard, contactless card, or mobile wallet. For additional passengers traveling in your group, a second Metrocard, contactless card, or mobile-wallet device would be needed for passengers 5-8. Free transfers within two hours are available from bus to subway, subway to bus, and -- in most cases -- bus to bus. You could, for example, take the subway from NYU to Columbia, spend an hour touring Columbia, and then get a free transfer to a bus heading back downtown.

Here's some helpful advice about riding the subway:

https://new.mta.info/guides/riding-the-subway

Which airport do you plan to fly into: LGA, JFK, or EWR? LGA will be the closest to Manhattan.