I suppose my answer would be the same as many other Upstate New Yorker's. It's a family thing.
A fandom seeded in some of my earliest and best childhood memories, way back in the early 1980's.
When I close my eyes I can still smell the hint of Genesee Cream Ale, an aroma coming from the coffee can half full of chew spit and a mix of bengay & cleaning supplies that was the living room in my Grandparents house. I can feel the 40-grit sandpaper that was my Grandpa Skinner's stubble on my cheek and the sting from my older brothers high fives that were equal parts celebration and reminders that he was the "bigger" brother. Lastly, and this is what really made the Yankees stick as a part of these memories, I can hear that jovial voice that really brought the game to life for me and just made it fun to watch. That would be the eternally young, love of baseball, that poured out of Phil Rizzuto. I truly believe The Scooter's love of baseball had some influence in my life long love of the game. My Grandpa's love of the Yankees made them my team for life. My grandfather had to be put in a home in late '84, early '85 and passed away when I was 9, in 1986 . The majority of my memories of my Grandpa center around hanging out, watching Yankee games with him and my brother. Even though I didn't remember the last World Series, let alone playoff, games before the drought my loyalty never waned. Mattingly represented everything the Yankees stood for to me and Rizzuto was that familiar voice that always took me back to that living room in that 2nd floor apartment on Amsbury St., with Grandpa Skinner.
Now, as I share memories with my children I think of him often. Telling them stories of Jeter, Mattingly, Bernie, Randolph, Guidry etc, I think of all the stories he shared with us. He was born in 1900 so man, did he have some memories to pass on. While I don't remember most of the stories I do remember that he hated the "Bronx Zoo" clubhouse hoopla, thought "that man Babe Ruth was sent down by God himself to make sure Baseball would forever be America's past-time", Mantle and Munson were two of his favorites due to all the obvious pain they played through & Gehrig was also for never missing a game. Those things really meant something to a man that fought in 2 world wars and survived the Great Depression. As I pass this onto my kids I am doubly excited as It looks like they won't have to wait until well into their teens to see another group make their mark on the list of Great Yankee Dynasties.
Hopefully in the future they will all have a similar answer to the question "How did you become a Yankee fan?". One rooted in cherished memories that can best be summed up in 4 simple words: "It's a family thing".