No, most America doesn't "know it" When I go to an A's game, no one is talking about the Yankees/Red Sox, nobody cares.
The rivalry appeals to Yankees fans, Redsox Fans and the causual baseball fans that don't even watch baseball until the playoffs start. Other then that, you'd be damn suprised at just how much people don't give a flying fuck. Sure as a baseball fan I think the playoff series between them will be heated and exciting, but the rest of the season I quite frankly don't care about their rivalry as I have my own team and their success or lack thereof to worry about.