The bill was created and pushed through the legislature by "Big Tim" Sullivan, a crime syndicate leader and politician who was part of Tammany Hall. His toughs had complained about immigrants who were resisting their extortion efforts. Big Tim had a simple solution. Make it illegal for his opposition to legally have or carry weapons. From the New York Post:
Sullivan knew the gangs would flout the law, but appearances were more important than results. Young toughs took to sewing the pockets of their coats shut, so that cops couldn’t plant firearms on them, and many gangsters stashed their weapons inside their girlfriends’ “bird cages” — wire-mesh fashion contraptions around which women would wind their hair.
Ordinary citizens, on the other hand, were disarmed, which solved another problem: Gangsters had been bitterly complaining to Tammany that their victims sometimes shot back at them.
So gang violence didn’t drop under the Sullivan Act — and really took off after the passage of Prohibition in 1920. Spectacular gangland rubouts — like the 1932 machine-gunning of “Mad Dog” Coll in a drugstore phone booth on 23rd Street — became the norm.