Museum's own FAQ says to ask a Public Safety Officer about specific gallery access for strollers; they also run dedicated Stroller Tours for ages 0-2.
They claim there's a nursing room, but had a hard time finding it while I was there; there is a "family room" near the entrance though. I was able to nurse on a bench in the museum. Changing table in many bathrooms
Brooklyn Museum
Prospect Heights, Brooklyn
Standard strollers are allowed in most galleries — just not oversize or jogging strollers.