Good topic!
It influenced me in ways already echoed by some others: I throw out lines with a few friends who know what I'm talking about. I also use it as a barometer for other films. I like to reference Hyman Roth.
The first time I saw a GF film was in 2001. I saw the last 1/2 of GF2 on USA Network. I was intrigued. From then, I saw GF1 in bits & pieces on either Bravo or AMC (they were showing it frequently). Once I saw GF1 in its entirety, I was blown away. It is my favorite movie of all-time, bar none.
I can't really say that it sparked my interest in the mafia because I had been reading about La Cosa Nostra for the last ten years and watching mob-related movies: Casino, Goodfellas, & Donnie Brasco. I read books about Gotti, Gravanno, Big Paul Castellano, Brasco, etc.
But for me, the acting in GF1 is so undeniably great. Brando was already a legend before the film, but GF made stars out of Pacino, DeNiro, & Duvall.