First, try tapping the bowls with fuel on, using the plastic handle of a screwdriver is good. If the float valves are just gummed up that might free them to seal.
If not: you will have to remove the carbs and their fuel bowls. This is a good idea regardless... if the float valves are gummed up there's probably lots more crud in there.
Just removing and cleaning the float valves usually brings them back. If they have brass needles, you can lap them into their seats with a spot of brasso, a but of plastic tube or wire insulation can be used to hold the needle and turn it in the seat. Rubber tip needles, or really corroded seats or metal needles need to be replaced. Use Honda OEM parts. Honda sells float valves and gasket kits, they cost more than aftermarket "carb kits" but they're genuine Keihin parts and worth it.
You should also clean out all the jets, emulsion tube holes (on pilot jet and the main emulsion tube), internal carb passages, and the bowl and carb inside. Be careful removing the float valve pins: they get corroded in, and using much force will break off the support post which is a pretty nasty thing to fix.
There are lots of forum posts on these jobs. Check in the "tips and tricks" topic.
