The fact that Moriarty has become shorthand for one's nemesis, I had to vote Sherlock Holmes/Moriarty, but Batman/Joker was a very close second.
You make a good point; for a great length of time in the previous century, Holmes and Moriarty were the cultural / literary go-to reference to the example of the great hero/villain pairing. As film and TV took a greater hold on the culture, Holmes and Moriarty began to take a back seat to the
few true superhero/villain pairings that transcended their medium of origin, such as Batman and The Joker, which had risen to new heights several times across the lives of characters in print, TV and film, not to mention a
consistent presence in merchandising since the 1960s, forever selling them as the very image of good vs evil.
After Batman and The Joker,
many rungs down the ladder might be Spider-Man and the (original) Green Goblin, since the Goblin/Osborn storylines (and being Spider-Man most personal, deadly enemy) were so essential in building the popularity of the hero into a major media player starting with the Romita/Lee period. Captain America and the Red Skull were once a popular hero/villain paring in the 1940s, but the decline of superhero comics (Golden Age) ended that from rising to level of DC's characters, even after the Skull was revived in numerous, great new and WW2 retconned stories starting in the 1960s and running well into the 1980s.
