A great classic site in those pre-YouTube days that I really enjoyed right as nostalgia for that decade was burgeoning.
https://web.archive.org/web/*/www.80stvthemes.com
https://web.archive.org/web/20040419141802/http://www.80stvthemes.com:80/Of course, YouTube and other video sharing sites provide most of what was provided here, but it seemed so magical back then just listening to the theme since I was born too late to record TV from the '80s, instead of relying on what reruns came on.
If you look below, you'll see that the main menu bar to the other parts of the site is still intact from the link above. One of those includes the game shows page.
Of course, I don't think the 1997 Jeopardy! theme really has an '80s flavor to it, only the 1984 and 1992 remix do, even though it's posted. It also includes some '70s themes as well like Tattletales, Second Chance, The Better Sex, and prize music in the '70s Dating/Newlywed Game versions. And the link for The Price Is Right page still works, though the downloads no longer do unfortunately.