petcock is #4 in this pic:
http://partsfinder.onlinemicrofiche.com/atlanticcycleandpower/Honda_OEM/HondaMC.asp?Type=18&A=58&B=57&Action=O
you should be able to switch between ON and RESERVE with this lever.
sure sounds like its starving for fuel as time went on so something is gumming up,
i would check to see if the fuel filter is clogged (#2 in the above pic) and if you can reach it, disconnect the output of the filter and see how it flows. if its a good stream, then focus on the carbs. if it doesn't, then replace it.
carbs may have gum & tarnish in them. you'll have to remove them and give it a thorough cleaning. if you're not comfortable doing any of this, then take it to a dealer or independant shop. if you're ok with removing the carbs, but not comfortable cleaning them internally, then take them to a shop that can do this.
if you can get them off and want to see just how bad they are, then remove the bowls (#5 in this pic):
http://partsfinder.onlinemicrofiche.com/atlanticcycleandpower/Honda_OEM/HondaMC.asp?Type=18&A=58&B=33&Action=O
and see how bad the gum looks. it will smell like varnish and look either very sticky or very hard. be careful removing the bowls to keep the gasket in one piece. it it tears, then you'll need a new one from honda.
pay particular attention to the jets (#19 & #20 in the pic) as these little holes plug up almost immediately once gas starts going bad. you can remove the main jet and slow jet and give them a cleaning with carb cleaner and a bread tie (with the plastic removed, using it like a pipe cleaner). they are brass so be sure not to deform them with tools.
going any further will take patiences and a little skill, but you can try the above at a minimum and put it back on to see if it will run.
good luck.
pic of your carbs as an assembly on the bike:
http://partsfinder.onlinemicrofiche.com/atlanticcycleandpower/Honda_OEM/HondaMC.asp?Type=18&A=58&B=32&Action=O