The Los Angeles Public Library was established in 1872. According to Ludwig Louis Salvator's 1929 book, Los Angeles in the Sunny Seventies: A Flower From the Golden Land, the membership fee to join was $250, plus 50 cents per month or $5 per year. Members were allowed to take out two volumes at a time. If the membership fee was still charged today it would the equivalent of $6000 to join, and $125 per year dues. Thankfully they went 'free' in 1891.
Bernadette Dominique Sotter's 1993 book, The Light of Learning: An Illustrated History of the Los Angeles Public Library,lists three earlier attempts at libraries in Los Angeles. In 1844, a group known as Los Amigos del Pais opened Amigos Hall, a dance hall with a reading room on the side. Donated books and magazines could be read there for free. It wasn't long before funding gave out. In 1856 it was the Mechanic's Institute that opened a small reading room which lasted until 1858. A Library Association was formed in 1859 and opened a reading room in the Arcadia Block. There was a $5 initiation fee and dues were $1 per month. The money collected funded the library and assured it a longer life than the previous efforts by Los Amigos del Pais and the Mechanic's Institute. That was until the Civil War killed it. Read all about it at a library near you!
**UPDATE: I think there was a typo in Salvator's Los Angeles in the Sunny Seventies, other sources cite the membership fee as $2.50.