From the Record Gazette newspaper:
A hundred years ago, in 1916, the Banning Library started as an idea of Robert Blee, Banning Union High School’s principal. Now, it has grown with the times, fitting to the needs of the community.
In honor of this journey, the Banning Library held a Centennial Celebration on Oct. 22. 
From email correspondence with the Banning Library District librarian:
An election was held on 8/21/1916 and the vote was in favor of a library for Banning and Cabazon cities. The first book shipment was received in October, 1916.