How do you fix a leak that is coming from the upstairs shower drain into the basement ceiling tiles?

It depends on the origin of the leak. If it's just the drain you might be able to tighten some fittings. In many of the older homes, (20 years or more) there is a pan seal used on tile showers that keeps water from leaking past the drain. The pan used back then was basically a hard tar paper/rubber combination that failed over time and ends up leaking. The tar paper was folded such that it fit tightly into the corners and up the wall approximately 4 to 6 inches. If the pan sealer has failed, the only real option is to replace the shower since you'll need to pull the tile to get to the pan. The pan used today is vinyl and won't break down like the old ones did.