I have gas dripping from my air cleaner. 2002 883R has been sitting for 1.5 years.

get the carb serviced. Could be fuel overflow. small engine carbs are very finicky if left sitting for too long with fuel remaining in the carb.

Did you put stablizer in the gas before storing it? Does the fuel pump run for 2 seconds when the ign is turned ? Have you checked the air cleaner for debris? Do you have spark? You must have air/fuel/spark to start.

Safer is to use 'carburetor cleaner' which is also flammable but a little less dangerous than using gasoline.After sitting a long time, our modern gasoline gets tired; never allow something to sit so long without adding some 'fuel stabilizer.'If possible with your car, I would drain (siphon) the tired gas out and replace it with a few gallons of fresh.
Really tough starting problems can also be overcome with 'starting fluid' which is pure ether and very flammable.Any of these may require the removal of the engine air filter so the liquid gets into the intake manifold.
Stale gasoline can also gum up the fuel injectors which may have to be removed for service if none of the fluids causes the engine to fire.

If pilot lights and flame is robust and clear blue, and pilot goes out later, then some suspects can be excluded: air in gas line, low gas pressure, thermocouple is overtightened (finger-tight + 1/4 turn), thermocouple not positioned correctly in the flame, dirty pilot tube.
If flame is yellowish and small: then clean pilot tube and combustion parts.
http://waterheatertimer.org/How-to-install-gas-water-heater.html#clean
Remaining suspects can include: air in gas line, low gas pressure, bad gas control valve, tight house so household vent fans pull air down water heater vent that blows out pilot, hot attic with inadequate ventilation, water dripping down vent.

Open following links to read troubleshoot resources.
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot

you can try and remove the old gas the carburetor and perhaps the fuel filter are most likly fouled with gunk from the old gas. When gas sits, over time it forms a thick gummy deposit that can clog carburetors, and anything else. try fresh gas with some carburetor cleaner (NOT the kind in the spray can) mixed in. It may need to have the carburetor rebuilt or replaced.

If I understand "the petcock was in prime position and it eventually quit dripping."
Prime = on
Sound like the needle rubber was hard and after a while they soften up enough to cut off the fuel.
Keep an eye on it and ride safe.
Ken

Has the bike been sitting??? If the carbs are gummed up from sitting and the float and or float needle get stuck in the "add fuel" position it could cause this

Spray them w/ oven cleaner, put them in a plastic trash bag, and let them sit overnight. They should clean up w/ minimal scrubbig.

the problem is that your carbeurator is gummed from from sitting idle all winter with gas in it. Get an aerosol can of carbeurator cleaner, remove the air filter and give the air intake two squirts of carb. cleaner. Then start the mower. When it stalls again let it sit for an hour while the carbeurator cleaner works at dissolving some of that gum. Repeat the process until the mower runs again. Sometimes there is so much junk in the carb. that you have to remove it and soak it in cleaner for about three hours, and then rebuild the carb. with the proper little kit. hopefully you won't have to take this step.

Next fall let the mower run out of gas and stall then you won't have the gum in the carbeurator in the spring.