What are my options for fixing a laundry faucet?
Try tightening the packing nut (the hex part the stem goes through) a little. That's what it's for. Don't overdo it. The "packing" is the material that seals around the stem - the packing nut compresses that material. On a valve that is not used often, actuating the valve does commonly cause it to leak, as things have "set." Tighten a little, wait, tighten a little more, wait - try to sneak up on it, as the pressure is a balance between enough to seal and little enough that the valve turns easily.