Deciding what the best decade to live in New York City wasn't easy. The 1920s, before the Crash, must have been incredible. The 1950s, with its postwar prosperity, and the 1960s, with its burgeoning arts/theater scene, are good candidates as well, as are the resurgent 1990s (thank you, "Rent" and "Seinfeld"). But I don't think any of these decades would have topped New York in the 1970s.
First, New York became the center of the movie universe in the '70s. Think of any random movie scene involving New York, and more than likely a 1970s movie is popping into your head -- perhaps "Saturday Night Fever" or "The French Connection" or "Manhattan" or "Godspell" or "Warriors" or ... well, you get the ieda. Next was the theater scene, as strong as ever.
But what really gives the 1970s to New York was music. Disco and R&B centered itself around the Big Apple during the decade, but so did punk (The Clash and Sex Pistols notwithstanding). Even the Rolling Stones were singing about Central Park. Add some Bruce Springsteen, and you see how important New York was not just in the 1970s, but also to the whole history of rock 'n' roll.
If you watch "Saturday Night Live" episodes from the 1970s, you see the New York culture reflected in the comedy. "Barney Miller" and "Welcome Back Kotter" weren't videotaped in New York, but the opening and closing credits of both shows conjure up indelible images of the city at this time. Add Reggie Jackson and the Yankees, and you see why New York in the 1970s is almost irresistible.
I've never been to New York, but this is the Gotham decade I'd want to be dropped into if that technology became suddenly available. The city was debaucherous, energetic, a little bit scary, iconic and alive. In other words, it was everything New York was supposed to be.