As far as popular culture goes, Randolph Carter has never quite been onstage, and not nearly as much as Lovecraft’s Cthultu mythos (and related works) has.
As far as Carter himself, he has been featured in several movie retellings of his stories, mainly Jean-Paul Ouellette’s two Unnamable films:
The Unnamable, and
The Unnamable II – The Statement of Randolph Carter.
In the musical parody A Shoggoth on the Roof Carter makes many appearances. You can watch A Shoggoth on the Roof on its official site, or with this link.
Carter has also appeared in a handful of games as well, such as the card game MYTHOS, where he appeared as an ally card.