Does your bike have a vacuum operated fuel petcock? Starting in '79, the 1000's did. If so, the lever will have three positions, main, reserve and prime. If not, instead of prime it'll say off.
You don't say which carb leaks gas, but if you have a vacuum operated petcock, it's probably the diaphram and/or rubber gasket inside the petcock that's leaking. The vacuum line for the petcock comes from one of the inside carbs and if the petcock is leaking, the gas runs through the vacuum line and directly into the carb.
If it's the float valve in the carb, it wouldn't make any difference what position the petcock is in. If it's the petcock, that would explain why it doesn't leak on reserve.
I don't think Kawasaki sells the petcock repair kits anymore, but your dealer should be able to get one aftermarket. I've also seen them on ebay. Just make sure you get the correct one for your particular model of bike because different models, even of the same engine displacement used petcocks with different petcock lever configurations. The repair will probably be around $30-$35 and is easy to replace. Just be sure to initially smear a bit of oil on the flat side of the petcock where it contacts the rubber gasket so you don't tear the gasket the first time you turn it while still dry.