Although Disney advertizes BW as an entertainment area like DTD (and it is, kinda), it doesn't have buses like DTD does. You'd have to take a BW resort bus from somewhere.
From the WL, I would guess the fastest route would be taking the boat to the MK and hopping on the BW bus from there. This would only work during MK hours though. Or you could take the DTD bus from the WL and transfer to a BW bus at DTD. If you were feeling like a walk, you could also take the Epcot bus from WL and walk though Epcot out to the BW area.
There are other variations on this (like taking the TTC bus/monorail/Epcot route), but the general answer is you can't get directly from WL to BW.
I agree, boat to MK and catch a resort bus. If the Swan/Dolphin bus comes before the BW bus, take that and get off at the Swan. It only takes about 3 minutes to walk from the Swan to the Boardwalk, and the BW is the last stop on the BW/BC/YC route so it takes a little longer.
The walk from MGM (about 12 minutes) would probably be quicker than entering Epcot, walking all the way through FW AND WS, and walking from International Gateway to the Boardwalk. You also can take a boat from MGM to the Boardwalk.
I would definitely have to advise to take a taxi.... one evening we left Epcot to go to the boardwalk area, and after we spent some time in Jelly Rolls we needed to get a bus back to our hotel.... ( I think we were in Coronado Springs then) well, the bus service late at night is pretty nonexistent...... so we waited and waited...overall the wait and the transfer, plus the time spent almost ruined a perfect day...