Sewell P. Wright's "The Commander John Hanson" series (e.g. "The Forgotten Planet") features Special Patrol Service, the space-police arm of the mercantile space empire known as Interplanetary Alliance.
They were published in Astounding Stories, from 1930 to 1933, well before 1937 (the full list can be found in "Science-fiction: the Gernsback years : a complete coverage of the genre" By Everett Franklin Bleiler, Richard Bleiler).
Also, a minor mention of the existence of space police (though the police per se don't play major role, so this might not fit the spirit of your question though it fits the letter) is "Uncertainty" by John Campbell (published in Astounding Stories, 1936).
There are other minor mentions of the existence of space police (again not as main part of the story) in several other works published in the 30s, as can be seen by searching for "space police" in the Google Books version of "Science-fiction: the Gernsback years".
Edit. The ten stories in Sewell Peaslee Wright's John Hanson series are available at Project Gutenberg:
"The Forgotten Planet" from Astounding Stories of Super-Science, July 1930, etext here;
"The Terrible Tentacles of L-472", from Astounding Stories of Super-Science, September 1930, etext here;
"The Dark Side of Antri", from Astounding Stories of Super-Science, January 1931, etext here;
"The Ghost World", from Astounding Stories, April 1931, etext here;
"The Man from 2071", from Astounding Stories, June 1931, etext here;
"The God in the Box", from Astounding Stories, September 1931, etext here;
"The Terror from the Depths", from Astounding Stories, November 1931, etext here;
"Vampires of Space", from Astounding Stories, March 1932, etext here;
"Priestess of the Flame", from Astounding Stories, June 1932, etext here;
"The Death-Traps of FX-31", from Astounding Stories, March 1933, etext here.
He mentions various space war and space police stories he studied when planning the Lensman series. The list could be interesting. Even if the earliest story he mentions turns out to be after Edmund Hamilton's first interstellar patrol story, some of the stories could be interesting to those who like really old space operas.
– M. A. Golding Jan 13 '17 at 04:44