The trade paper Variety ran a story the day following the momentous premiere that made mention of the police presence that night. It read in part:
20,000 Jam Streets at Snow White Premiere
Crowd of more than 20,000 jammed entrance to premiere of Walt Disney's first all color feature-length cartoon 'Snow White and the Seven Dwarfs' at the Carthay Circle Theatre last night. Assistant Chief of Police George Allen had 80 coppers on hand, 15 more than for the ordinary preem [sic] detail. Aiding them were 11 RKO studio policemen. Two women fainted in the early crush.
Over the course of the following weeks, hundreds of newspapers across the nation devoted columns of text to Disney's achievement. Here's one such account, which was published in the Twin Falls Evening Times on December 23, 1937. The article's headline and slug read:
Disney Premiere Proves Success
Hollywood Celebrities Pay $5.50 Each to Witness Full-Length Cartoon
My friend Robert Lughai has also published a great post detailing premiere night on his own blog Filmic Light Snow White Archive - it's well worth a visit.