Emphasis on history: NYC very famously broke its organized crime groups in the 1980s and 1990s. It's what made Giuliani famous before he became a politician[1].
(I would hazard a demographic claim around organized crime: just about any mid-sized city with large suburbs almost certainly has more per-capita organized crime than NYC does. You just don't hear about it because most of it is of the "extortion for trash pickup" variety, not the "Murder, Inc." variety.)
I took pains to mention the extant nature of organized financial criminality which yet influences NYC (and state, and national) politics. Wall Street gets their way a lot when they shouldn't, and it's because government officials and elites are happy to pledge fealty to money over law.
As for Giuliani, he himself is a mobster; he's facing the same RICO charges he leveled at crime bosses as a prosecutor. I think this speaks to my point, which is that NYC corruption vis a vis organized crime didn't go away, it just became part of the institution.