Cold water shut off valve in basement, that cuts water supply to outside shower and hose hook ups, is dripping. I drained the outside pipes. I shut off the valve inside. Then when I shut off the outside hose faucet, the valve indoors drips. If I open the hose faucet outdoors (still leaving the valve shut inside) that drips. What's up?

Where's the valve leaking from? The spout? Under the handle at the stem? Let me know, Tom

From the spout oudoors.
From the small drain hole at the turn off.

from the spout oudoors.
From the small drain hole at the turn off. What's the small drain hole at the turn off? Is this a frost free hose bib?
If it's leaking from the spout you need to replace the washer. While you have the valve apart run your fingernail around the seat. There should be no nicks or rough spots on the seat, If there is the valve will have to be replaced. Good luck, Tom