I think Conrad is talking about Dumbo, or possibly Brooklyn Heights. I live in BH, and it's wonderful, although very expensive. The neighborhoods that are "established" and nice are the aforementioned two plus Park Slope. All are expensive. Williamsburg, Ft Greene, and Greenpoint are still in the process of gentrifying somewhat, but many young professionals live there and I gather the rents are slightly more reasonable. I've heard good things about Bay Ridge too, but it's very far out in Brooklyn and you probably would want a car.
The OB units I'm familiar with are both in Manhattan - New York Presbyterian Cornell, which is on the west side, is where I did my OB clinicals. It was gorgeous, but with very high epidural and C/S rates, and a certain sense of . . . entitlement? I guess? The pt pop tends to be primarily upper-class there. New York Presbyterian Columbia is on the east side, way uptown, and that's where I work, in the NICU. Both have great L&D with some cool high-risk stuff. If you want to stay in the boroughs, Long Island Jewish does good work. Keep in mind that I tend to judge L&D by the NICU - if the NICU is a level III, then there is generally high-risk antepartum work going on there, which is interesting to me, but may not be to everyone else.