New Haven for sure is not Princeton or Cambridge, but compared to the areas surrounding Columbia and UChicago, , even UPenn, it is more than a leg up on them.
Columbia?? This is not the NYC of the 1980’s; Morningside Heights is a perfectly fine neighborhood, IMO.
The area around Columbia is far far safer than New Haven, plus there is much more to do. Not even comparable.
I own a place in Morningside Heights, and while I would not say the surrounding area is safer, most students at Columbia do not venture past 125th and north, except for the occasional trip to Fairway Market on 132nd. Fact is, without being overtly stupid, I feel safe in both New Haven and in the neighborhoods surrounding Columbia.