i'll be attending downstate in june as well.
you seem to have a pretty good idea of where to move to in brooklyn, for park slope you will want north slope- near the 2 train (bergen street or grand army plaza stop). brooklyn heights is close to the trains although probably the furthest from the school of all those options, but not by much and its a really pretty area. prospect heights is nice too, a bit more up and coming but convenient to the 2 trains and a bit more affordable than park slope. fort greene is also a nice area and close to the 2 train at atlantic avenue. the commute should not be more than 30 minutes or so, but you can get exact info on www.hopstop.com where you can put in the start and end stations and it will tell you how to get there and the approximate time. the transport section of the student bulletin it explains how to get there and there seems to be some sort of a shuttle for evenings and non-rush hour times http://sls.downstate.edu/admissions/pdfs/bulletins/nursing_bulletin.pdf. let me know if you have any more specific questions about neighborhoods and good luck in your housing search.