If you live in a colder weather climate (though your picture indicates that you probably don't due to plants showing at this time of year, and it's not a freeze-proof sillcock), you may have an inside shutoff valve to that outside faucet. If you do have an inside shutoff valve, you could turn that off for a couple days and see if the dampness reduces or disappears. If it does, that would indicate that the faucet plumbing is the culprit. If not, look elsewhere.
Though the dampness all the way up to your siding would indicate that the faucet itself may not be the culprit, especially if you don't see any direct signs of water either on the inside or outside, and dampness might not typically run against gravity that far unless it's really flowing. Is there any plumbing in the walls above that location that might be leaking down? Bathrooms or showers/tubs that may be leaking into the floor and running out? A window that might allow water to get behind the siding and drip down to the foundation? Roof or gutter issue allowing water to run down behind the siding in that location?