If you have a neighborhood that you really want to live in but just can't afford, one easy way to find cheap apartments in NYC is to look in nearby neighborhoods that are somewhat less desirable but offer apartments at a significant discount. Here are a few examples:

Comments: If you want to live in the Greenwich Village area, the East Village is the cheapest neighborhood. It is a bit grungy, further from trains and louder in the evening. Alternatively, this means it has more cool bars and restaurants and a younger vibe.

Comments: Battery Park lacks the charm of Tribeca but offers great views of the Hudson River, more park space and full service buildings. Plus, Tribeca is only a short walk away. This is a great alternative for cheap apartments in NYC.

Comments: Park Slope South has great access to the stores and restaurants of 5th and 7th Avenues, but at a lower price than Park Slope as often defined by the 321 school district. If you aren't looking to send kids there, you should think about living just outside of it where you get the benefits of the neighborhood with a little bit less cost. Also, if you want to live north of Union Street, this is also a great neighborhood outside of the 321 school district.