Unless you want to write fiction "like" the work that was being written in the late 30's, it would probably be a better idea to research the police in New York as if you were doing a nonfiction work. I approached one of my books this way -- it was set in the 1930's in a major US city and was based loosely around a real case.
There are literally HUNDREDS of books on crimes in New York and within many of them you will find a lot of information about how the police department was run. For example: BAD SEEDS IN THE BIG APPLE: BANDITS, KILLERS & CHAOS IN NEW YORK CITY, 1920-1940, among many others.
Look for police officer biographies of the era. Look for big cases of that era. Always pay attention to the appendix, footnotes, etc. for clues as to sources.
Also, look at papers written for Criminal Justice classes at the big New York colleges -- often a thesis will tackle specific aspects of policing in a particular era.
I always study photo collections of the era I'm writing about so I get the look of the streets pictured clearly in my head.
Newspaper archives are fantastic for that sort of thing, too. And they often deal with police procedures -- especially when you can locate an in-depth piece, often a Sunday supplement.
Watch movies of the era, too -- likewise, there are a lot that depict New York police in the time frame you're interested.
But you need to pick a specific era -- New York in general might be just too big.