Rusted Agitator Shaft On A BD WP Washer
The quick way to fix this washer for a while is to do what Martin suggested, get another agitator that at least fits some what on the old rusted shaft, then with the agitator in place drill a 1/4" hole through the agitator and shaft and put a strong bolt through it, then drill a 2nd hole 3/4" above the other 180 degrees from the other and put a 2nd bolt through.
I have seen this done and if you do a good job it might last a good while.
Other than this you have to pull the transmission as Eugene suggested and replace the shaft, there is no need to try to rebuild the old shaft however, new shafts are all over Ebay for as little as $20
John