Ancestry released a collection today with almost 18,000,000 names of Manhattan Voters. These books were stored at the Municipal Archives and digitized over the last two years or so. You can search them at https://www.ancestry.com/search/collections/62363/.

Before 1957, voters in New York State had to register separately for each election, these registers are the annual registration registers and generally contain the following information: name, age, country of birth, date and place of naturalization, date of registration, occupation, place of employment, residence, party affiliation.

Not all books for all years survive, but if your ancestors regularly voted you may find 1 - 2 dozen records for them.

Note: This only covers Manhattan.
Queens and Staten Island records are available on site at 31 Chambers St. Brooklyn records are still with the NYC Board of Elections. The Bronx's voter registers have been largely destroyed through flooding.