The Brooklyn Bridge has the distinction of being the only major bridge in the city not designed for cars. It was completed in 1883, two years before Karl Benz built the first passenger car, and 25 years before the Model T Ford. For city dwellers, a private horse-drawn carriage was a definite luxury, and most people crossed the bridge by train, trolley or horsecar. When the bridge opened it had one heavy-rail track, a trolley track, and one lane for carriages, in each direction, plus the pedestrian walkway. In 1944 the BRT (that's Brooklyn Rapid Transit) trains stopped running, and in 1954 the trolleys went out of business, leaving the city to devote six lanes to private cars and taxis.
With the increasing popularity of cycling in the city and in Brooklyn, there is often tension between cyclists and pedestrians on the walkway. The North/East path of the parallel Manhattan Bridge is primarily for cycling, but this does not seem to have reduced demand for the Brooklyn Bridge path.
The bridge currently does not allow commercial traffic or buses. In this post I assume that it is structurally capable of handling bus traffic, since it was designed for heavy rail, but it may have worn out since then. If it turns out that it is not safe for buses, then it should be redesigned to fully accommodate the existing demand among pedestrians and cyclists.
As with the Brooklyn Battery Tunnel, the Brooklyn Bridge has a transitway that can feed it: the eight-block bus-only Fulton Mall. Five bus lines use the Fulton Mall for most of its length, and five others run on Livingston Street, one block south. These are all local buses connecting to nearby parts of Brooklyn, but some of them go as far as Long Island City, Ridgewood, East New York and Marine Park. Many of the people using these buses come to work in some of the offices, colleges and courthouses of Downtown Brooklyn, but many change for the subway to Manhattan. If we sent some of these buses over the bridge, a lot of them wouldn't have to change.
Between the Fulton Mall and the entrance to the Brooklyn Bridge are the three blocks of Adams Street, one of the most over-designed streets in the borough. Adams Street can easily spare two lanes for dedicated busways. Those lanes can be connected to dedicated busways on the bridge itself, using the inner lanes of the roadway just like the old BRT tracks - BRT returns to the Brooklyn Bridge! If we really wanted to get creative, we could dedicate one or both of the outer lanes to bicycles so that pedestrians can have the walkway to themselves.
In Lower Manhattan, just about everything is a block or two from everything else, and if - as described in the previous posts - we've built dedicated busways on Sixth Avenue, West Street and/or the FDR Drive, it's a relatively simple matter to connect them to the Brooklyn Bridge. Many of the local buses that now terminate in Lower Manhattan could go to Downtown Brooklyn, or through-run to spread out destinations in Brooklyn, and vice versa.
At the other end of the Fulton Mall, many buses get stuck in traffic on Flatbush Avenue, which is a real bottleneck. The overall state of bus travel in Brooklyn is not going to get significantly better without removing that bottleneck by dedicating at least one lane of Flatbush in each direction to buses between Fulton Street and Atlantic Avenue.
So there you have it: to have Bus Rapid Transit over the Brooklyn Bridge, you first have to allow buses on the bridge. Then it's a relatively simple matter of running the Fulton Mall and Livingston Street buses down Adams Street, and figuring out where they go once they get to Manhattan.
Simple, yes. Easy - especially politically? Not so much. But all these posts assume a certain level of political and financial support for BRT. Without that, you're not going to get much BRT anywhere in the city.