How to approach fixing this kitchen sink faucet leak at the base
It is difficult to determine the source of the leak from your pictures, water leaking down the chrome faucet body is almost impossible to see without careful scrutiny; it could be coming from one or more of several components.
I will address a likely cause and help you on your way to disassembly and troubleshooting.
When the spout swings around it does so while riding on two rubber o-rings. Water enters the spout from the main mixing valve and these o-rings prevent that water from leaving the spout except for at the sprayer head. The o-rings get worn flat over time, and the seal is compromised, they will be located near the top and bottom of the spout body:

To remove the spout and change the o-rings, you must remove the handle and the retaining cap. Start by using a small flat screwdriver to remove the trim plug that hides the handle retention screw:

There will likely be a set screw in there, possibly requiring a hex wrench. Remove the handle, the retaining cap, then pull up on the spout to remove it. I would suggest replacing all seals, o-rings, valve cartridge at this time. And get some waterproof plumber's grease, it will improve longevity of the spout o-rings and make your spout swivel nicely.