Diagnosis
The leak isn't from the RO system's plumbing, it's from the installation hardware of the faucet itself. The mounting nut underneath the counter is probably loose or the seal/washer is compromised.
How to Fix It
You'll need to get under the sink. Hold the faucet steady from above while you tighten the large plastic mounting nut underneath the countertop. Don't overtighten, just snug it up. If that doesn't stop it, you'll need to disassemble it. Remove the nut, take the faucet off, and check the rubber or plastic gasket/washer that sits against the countertop. Replace it if it's cracked or flattened.
If That Didn't Work
The leak might actually be coming from the tubing connection at the base of the faucet and running down the stem, making it look like a base leak. Check the compression fitting where the 1/4" tube connects to the faucet valve underneath and tighten it slightly.
Heads Up
If you have a soap dispenser hole or air gap faucet, make sure the leak isn't actually from a different hole in the sink where water is splashing. Isolate the leak by drying everything and running only the RO faucet.