Coming to NYC from just about anywhere else in the country will be a shock. I've got a friend from the mid-west who is basically Larry the Cable Guy and whenever I've talked with him about coming to NYC there seems to be a general concern (I'd almost say fear in his gut) of coming here. Too many people. Too busy. Too easy to get "run over" (hypothetically and literally) compared to the easy country roads of where he's from. Heck, I had the reverse culture shock going out there for the first time in my life for college.
But the key thing from what I just said is "too many people". In a city like New York, you can't have 8 million people PLUS visitors and not have that many cops. Sure, you did see one on every corner, but was that in Times Square? Financial District? Some other tourist area? As well, some of those might just be the parking meter or traffic staff which typically don't carry a weapon, only a billy club, radio and maybe handcuffs / zip-ties at best.
Not saying it doesn't look like a police state, but as someone who has spent most of his life on this side of the country, I'd say it doesn't really bother you and when something bad does happen, you're glad there are these patrols around to respond quickly. A city the size of NYC would fall into chaos if they weren't there. In general the police aren't here to mess with you and they've got bigger things to worry about. When I was in high school, I remember drinking a beer on the sidewalk and asking a cop a question and there was no problem (yes, that would be underage btw). They're just doing their job but when the order comes to preserve order, they need to be tough because the city is a tough place. Heck, the city is safer now than it has been in many decades.