It could be the fuel line and it could be the automatic fuel shut-off valve is faulty. Between the tank and carb is a spring loaded diaphragm valve that is operated via engine vacuum. It opens when you start the engine and closes when you stop the engine. The float valve inside the carb is too weak to overcome the fuel pressure in the line by itself so the fuel valve cuts off the flow of fuel into the carb. If the valve goes bad then the carb can overflow slowly when the scooter is parked.
Most 50's have a valve that bolts to the tank with a strainer that projects up into the tank like the ones pictured below. To test you remove the fuel line from the carb and see if fuel runs out of the line. It should not flow out freely. There will be a little bit of fuel in the line but that's all that should run out. Sucking on the vacuum line will open the valve.


Sometimes the valve is bolted to the frame:
