Safest AND Cheapest places to live as close to Manhattan as possible?

I am planning on moving to New York within the next year. I am young and have never lived in a large city before so I wanted to make sure I was in a safer neighborhood. I know Manhattan apartments are out of my budget, but I’d love to be as close as possible. My budget is cheap…no more than 1200 a month and in a cleaner, safer neighborhood close to Manhattan…is this even possible??