If you leave campus housing, you can pretty much count on never getting back in. That being said, search at your own risk. It’s not impossible to find a cheaper apartment outside of NYU; however, usually there are compromises to be made. Affordable off-campus apartments can be far away and are usually not nearly as spacious as some NYU has to offer. They also require signing a yearly lease, so if you don’t plan on living in the city year-round or subletting to someone else while you’re away for the summer, then an off-campus apartment is not the right choice.
NYU dorms are pricey, but there are actually city apartments that can be cheaper. Craigslist is a great website to scout for places—a lot of times, you can get hidden steals. Another option is to see apartments through brokers, but watch out because some of them can be pretty sneaky. NYU also has an Off-Campus Housing Office (OCHO) that does a great job helping students find roommates and apartments. There are sacrifices to living off campus, such as a longer commute to class, not always having a security guard, or walking up apartment steps (no elevators!). You may even have to (gasp!) move to Brooklyn to afford housing. It's not advisable to move off campus as an underclassman because you will find a lot of your social circle will be made in your first few semesters. Suggestion: wait until junior year to move off campus and choose a few friends you will be comfortable living with. Many landlords are willing to negotiate with students, so finding off-campus housing may still pay off instead of dorming. Finding roommates with whom to share the rent doesn't hurt, either!