I honestly would say Dodgers vs Giants, because they have been more evenly matched overall. A lot of the history between the Yankees and Red Sox is of the Yankees just beating up on the Red Sox.
Of course it's the Giants and Dodgers. Anyone who says "Yankees and Red Sox" is simply saying so because we've been told over and over that they're the best rivalry. But when you look at the facts, there is truly no comparison. Giants and Dodgers have been rivals 27 years longer than the Yankees and Sox. Giants and Dodgers have played over 250 more head to head games against each other than the Yankees and Sox. The Yankees have a 200 game lead in head to head matches over the Sox. While the Giants have a 15 game lead in head to head matches against the Dodgers. The Yankees and Red Sox rarely ever effect each others seasons. On the contrary, no two teams have more first and second place finishes or have knocked each other out of the wild card as the Giants and Dodgers.
Although a good rivalry, the Yankees and Red Sox is largely a result of East Coast media bias. There is no measurable category in which anyone can say that the Yankees and Sox are the better rivalry. And if you want to get into fan behavior during these rivalry games- no opposing fan is intimidated to the extent that Giants and Dodgers fans are during rivalry games in opposing stadiums.
I would say Yankees vs Red Sox, the games are always a must see.
I would like to add Yankees vs Mets into the mix. Having lived in NYC most of my life and growing up a Yankee fan these games are electric.
The 2000 Subway Series was awesome.
Yanks and Sox
Dodgers vs. Giants. Yankees and Red Sox rivalry haven't been intense in a while, because the Yankees aren't a playoff team.
Yankees and Red Sox.
Dodgers vs. Giants, it actually started in NY in the early 30's when a Dodgers pitcher threw at a Giants batter because the Giants pitcher was laughing at the Dodgers, so the story in the paper went, I believe it was 1932 in Ebbets Field..