i'm doing this from memory -dont have a picture in front of me. I think the float bowl is leaking fuel and it runs onto the butterfly shaft and then runs off the ends of the shaft, problem is how it gets to the shaft in the first place.
Old carb's shafts are usually brass and will wear with age- but the usual problem with worn shafts is they will leak vacuum, not fuel because any fuel hitting them will be sucked down the intake when motor is running. But you can open the trottle and grab the butterfly and move it around while looking at the end of the shaft- you will be able to see if there is play. I have drilled out the shaft hole in the cast iron and hammered/ epoxied in a brass tube of the proper id to fix worn out cast iron body, if the shaft (most likely since its the softer metal) is where wear is (remove the butterfly and then pull out the shaft-looking for a "stepped" area where the shaft touched the cast iron) then you must replace or fabricate a new shaft, or find a lesser worn shaft from another carb.
if the fuel level is too high (from a non seating float needle/seat valve) then at idle raw fuel can spill down into the venturi area of the carb. At idle the fuel spillage maybe enough to run out the throttle shaft, but it would have a terribly rich idle with a good deal of eye-watering black smoke out the tailpipes.
Just ideas off the top of my head at the moment, will try to find good model BC picture in my Rochester carb book by Doug Roe. He had good words/pictures of this old carb (a little strange by todays-oops-later designs).
