After the blossoms fall off the Hosta plants at the end of sumer, should I cut down the stems or leave them alone?

I'd always cut them off. I don't think very many Hostas make seeds, so that would be the only reason to leave the stems, if you see seed pods.

I leave the stalks on just because I get some volunteers! Some cut them before they bloom just because they don't like the stalks, really up to you and your preference. ViolaAnn is acknowledgeable on all things Hosta!