I'm a lot more familiar with the West Side tunnels than I am with the East Side bridges, but I'm pretty sure that on the Brooklyn Bridge, as with most major bridges, there is little to no shoulder. Just browsed through a bunch of online pics to double-check, and now I'm even more sure.
If you have an emergency, the best option is to get off the bridge if you can make it. If not, try to pull to the side. If you can't do that, stay put. If you do stop on the bridge, people will go around you. Emergency vehicles will make their way over (once someone notices or a diver alerts them), as appropriate. I think there are usually a couple of city tow trucks assigned to the bridge, waiting at one or both ends, so if anything happens, they can clear the way quickly. They won't shut down the whole lane if there's any choice about the matter. They'll just direct people around the problem (closing off a relatively small section of the lane) until it's cleared.
Paul