New Yorkers themselves aren’t necessarily targeted for pickpocketing, but tourists are more likely to be. Nothing compared to parts of Europe, but more so than in the rest of the U.S., because density makes it possible. By contrast in most U.S. cities, we don’t usually get close enough to people to get pickpocketed.
The pushing on the subway tracks has happened occasionally over the years, but of course doesn’t happen frequently enough that they’ve built preventative measures, as some cities have for subway track suicide jumpers. But personally, I always stand with my back to a wall when waiting for the subway anywhere, as a safety measure in general.
There’s usual crime, like someone I know had their purse stolen in NYC while trying on clothes at a department store — someone lifted it from a hook in the dressing room while she was unaware. But that sort of thing can happen anywhere if you don’t mind your stuff, whether in restaurants, public toilets or such. I’d consider that a standard precaution in any city.