kitchen faucet leaking from swivel
These usually come apart by first removing the handle. Look for a set screw or cap at the end of the handle base and remove it. Now the cartridge will be exposed. There's usually a large nut or threaded collar that holds it in place. Remove the nut/collar and you should be able to remove the cartridge. You should then be able to see the bottom of the swivel and locate some sort or locking ring. Once removed, you should be able to pull the swivel out, exposing some "o"rings. You would want to replace them. 

Even if everything comes apart, there's a good chance the new "o"rings won't solve the leak. If you look closely at the top where the swivel meets the base, you'll see a lot of pitting/corrosion. If that corrosion goes down to where the "o"rings sit, you could be out of luck. It might be time to decide if you really like that faucet or if there's another type you'd like more and replace it.