I use suction cups, like the kind you use to keep your heater from touching your tank. I just run a piece of air tubing in there. I don't use an air stone. I have seen people using a clothes pin to keep the air line where they want it. I just tap off that.
Some people use powerheads, or small air pumps.
When using air it is best to have a valve on your line, so you can control it. The eggs don't have to move very far, you want them to just bounce gently. They say never take the eggs out of the water. Sometimes I become lazy and just dump them in there. When I am being careful I use a turkey baster, I usually strip the female in a small plastic tub, and then suck the eggs up in a turkey baster and put them in to the hatcher.