How do I fix this?

Get me the model and I can help more but this sounds like a vibrating washer. Is it more when using cold or more using hot?

Only when using cold

It's a Delta leland single handle pull down.

This diagram should be your faucet or close to it, no washer here. Remove the valve and check for debris by blowing backwards through the valve. Product Documentation : Customer Support : Delta Faucet (http://www.deltafaucet.com/customersupport/documentation/9178-DST.html?document=pd)

Possible loose washer in stop valve on cold side. Possible debris in cold water side of faucet or supply. Replace cold water stop valve. While you have the cold water supply disconnected, point in into a bucket. Open faucet in mid range between hot and cold. Hot water should flow out of cold water supply and flush faucet.

If replacing stop valve does not correct problem you will have to remove cartridge and flush faucet.

Thank you all! Will give it a try.