Prospect Park is surrounded by these neighborhoods: Windsor Terrace, Park Slope, Prospect Heights, Crown Heights, Prospect Lefferts Gardens, and Prospect Park South.
You can see that from this map
City of Neighborhoods - New York City Department of City Planning
It's an official nyc map from the gov't but I really think they are wrong by showing that Crown Heights does not border the park. It does, and it north of Prospect Lefferts Gardens and south of Prospect Heights.
Anyway, the "good" neighborhoods near the park are Park Slope and Windsor Terrace. Kensington is nearby as well and also nice. Prospect Heights is hip and also sort of expensive but IMHO is not safe enough to merit having those prices. Crown Heights is not a nice place to live (my opinion only). Lefferts Gardens and Prospect Park South are considered sketchy by some and have the low-ish rents to match but plenty of people live there successfully and safely.
Having said all that, I actually think you are going at this all wrong - which I think may be what seventhfloor was getting at.
First, you determine what your price range is for monthly rent. Will you consider roommates? This will help if you are on a budget. Remember to factor in utilities. THEN and only then do you move on to figuring out what neighborhood is best for you.
And if you are truly on a budget, you may want to expand your search to areas of Brooklyn that are not near the park, and also to other boroughs such as Queens, which tend to be cheaper.