This old fashion drinking fountain leaks. How do I fix it?
That looks like a Halsey Taylor "bracket style" fountain

You could contact them for repair parts availability, or call plumbing supply houses now that you have a manufacturer and model.

If you open the installation instructions, it provides a parts list that includes 


  600116618640 Stem Assy W/Lever


I googled that part number, and found this image at amazon:


And that looks like there is a gasket at the threaded collar that goes into the body of the housing, and a simple flat washer at the end of the stem. I'd replace the flat washer, and clean it out well first.