Back then, the people who went to these games actually lived in the city. Even the Dodgers of the '50s, for the most part, lived in Brooklyn. The subway was how you moved around. Most of the fans arriving at Yankee Stadium or Shea for the World Series are likely to do it via automobiles, do it from suburban points of origin in Long Island, Connecticut, Westchester County and New Jersey. You pay those kind of prices, you don't ride the subway very often.
Media makes it bigger.
Baseball is more than just a game now. To romanticize otherwise is to ignore how each team reached this point. What is not different is the emotions that will be stirred over the next 10 days. Back in the day, the late Dodger Pee Wee Reese recalled, he would walk into a restaurant with his wife and someone would issue a good-natured heckle. You're not as good as Rizzuto.