More than likely some grit or build-up is stopping the float bowl needle from stopping the fuel flow through the needle valve seat.
To test and clean, turn off fuel tap and drain fuel from carby via drain plug. Loosen carby manifold clamps and rotate carby enough to undo floatbowl. Return carby to level position and place a spill container underneath carby, turn fuel tap on and off (enough to allow small flow of fuel through open needle seat).
Now, gently hold up floatbowl with enough lift to close off needle valve seat and turn fuel tap on. If fuel still comes out you have a severe needle and seat problem requiring a replacement, if only a small weep of fuel is emitted it may just need a clean with a carby cleaning spray.
If no fuel comes out, your problem is the height adlustment of the floatbowl's level. Your engine is not using all the fuel that can be delivered and flows out the overflow outlet. The floatbowl level should be re-set incrementally to avoid leaning out your engine at high revs with too low a float level. To close off with a lower level you need to bend the small tab upwards.
The link below is for the same carby in exploded view format with all parts named. Cheers!
http://www.cmsnl.com/yamaha-pw80r-2003_model11164/partslist/A-09.html
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