My exterior faucet leaks around the stem only after opening the valve. It does not leak after closing. How to I stop a stem leak?

There should be a packing nut behind the handle, just snug it up until it stops dripping. If you tighten it too much you will not be able to turn the handle.

I am not sure which one is yours out door spigot they are almost the same see the images below, do as Puffmugs says if that will not work then remove the packing nut and replace the packing, Good luck.

John

Sounds like you already have your answer. :)