Let's see, leaks when running water from faucet, and stops when you quit running water from faucet. Could the leak perhaps be somewhere from the sink drain connection?? Are you connected to shore water or using water pump? Either way if the leak is from the fresh water supply source, it should be leaking after you shut faucet off as pressure should still be in the line. Or as another suggested, if you have one of those pull-out faucets, I would check where the flex hose connects to the water supply source.
Thecrew wrote: If you have an icemaker, I would check that line,valve and connections. Sometime the icemaker hookup is under the sink. Good luck.
As stated above, if it only leaks when the water faucet is turned on then that takes the water lines out of the picture even if it does have an ice maker because the lines will always have pressure on them. That only leaves the faucet.
Check the waste lines also check the clean-out cap may not be screw in all the way and at times the slip joints work loose just tighten with channel-locks and stutor vent
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Aside from a pull out faucet line your shut off valve could be leaking when on and not when off.....I think a new kitchen faucet would solve your problems....or a rebuild kit if you would rather go that route.