Did a bunch of rotations there as a med student. So first off let me say from the bottom of my heart, may god have mercy on your soul.
As far as housing, some of the residents live in the apartment complex directly across from the hospital. Otherwise, you don't want to live anywhere near the hospital. Brownsville is one of the most impoverished and violent parts of NYC. I would look for housing in some of the nicer areas of Brooklyn and plan to have a ~20-30 minute commute.