I have a Grohe Lady Lux 2. The cold water leaks down the inlet hose under the sink when I run the cold water only. Hot water is not affected. No leak when water is not running.
It sounds like you may have a seal that came loose or is broken. If you are handy, you can remove the faucet handle and you will see the seals. We had this happen to us once and it turned out that there was a loose screw inside that had worked its way loose. I tightened the screw and put the faucet handle back together and it worked fine.
Does it only leak WHEN you run the faucet? Because if it were a leak in the inlet hose or where it attaches to the faucet, it would leak all of the time, not just when the faucet is running. Then if it were a leak in the faucet itself it would likely happen when you ran cold or hot water. The details matter.
And how much "leakage" are we talking about here? For example it might be condensation running down the inlet pipe because the movement of cold water through the pipe causes moisture in the air to condense on the outside of it and drip down.