Manhattan is saturated with police as well as integrated with a web of linked surveillance cameras. Brooklyn has fewer but still far, far more then other parts of the country, especially out west. New York City has roughly 50,000 police officers and a police budget of over $4 billion dollars just for the city alone. On top of that you have state troopers and a variety of other agencies (like the MTA, the many federal police agencies in the city and the heavily armed army/national guard troops that are often visible in uniform/armor at Penn Station and other places around the city) who have their own police officers. Within 30 miles of the city you have thousands and thousands more police officers from Nassau, Westchester, Suffolk, and other surrounding counties (as well as additional police forces from towns within these counties). You're talking about 70,000+ officers within a 50 mile radius.
By comparison, Arizona has a total of about 14,000 police - for the whole state. Colorodo has about 12,000. Most states are comparable to this. You are just far, far less likely to see police officers in most other places in the country. As a New Yorker, I'm always astonished by the lack of police presence in other states (let alone other countries) because policing is so ubiquitous here.
I went to a Yankees game, saw Blackbird with Jeff Daniels and Michelle Williams (forget the theater), Smalls Jazz Club in the West Village, Nom Wah Tea Palor in the Chinese district, Diwanekhaas in the financial district, walked the Brooklyn Bridge then around Brooklyn. I mention the different locations and where I went because I walked everywhere to and from then around if something caught my eye. The only time I didn't walk is when I took the subway from Smalls Jazz club was back to the apartment I was staying at which was near W 48th and 8th.
As I stated that was my first time visiting, so I am not really sure how that covers NYC from a local perspective - I was certainly tired after 5 days. However, I did see police everywhere I went, except at 2am when I was walking from the subway back to the apartment.
Every place you mentioned I would expect significant police presence either because of the venue or locale. Since the Germans 'artists' swapped out the flags atop the Brooklyn Bridge there has been a permanent and rather disrupting police present on the bridge itself.
You realize those areas are 'New Brooklyn', recently came into money. Its not a coincidence that they are close to Manhattan and get increased police force for the rich yuppies.
Source: born in and lived in brooklyn for over 20 yrs.
I would complain that I don’t see enough cops on the street (live in Brooklyn, work in Manhattan)