Since Monday, the Brooklyn Public Library averaged over 900 library applications per day. Through Thursday, the Queens Library received an influx of 1,800 sign-ups, a 34 percent increase from the previous week, Elisabeth de Bourbon, a spokeswoman said.

By Friday morning, more than 9,500 tickets had been reserved on the Culture Pass website, culturepass.nyc, Josh Balber, a spokesman for the program, said Friday. With Culture Pass, those who have a valid New York City library card can reserve spots online for two to four people at any of the attractions. A total of 58,000 passes will be available from the arts organizations for the year, a number that may grow, Ms. Montefinise said. For example, 50 passes to the Frick Collection had originally been available; the Guggenheim initially had 100 passes available per month, but because of the popularity of the program, the museum decided to increase the number to 200 passes a month, a spokeswoman said.

The passes are released monthly and are available for dates up to two months in advance. Cardholders may only visit each institution once a year. The next wave will be available on the Culture Pass website on Aug. 1.

Passes for the Whitney Museum of American Art, Wave Hill, the New York Transit Museum, the Morgan Library & Museum, the Met, the Intrepid Sea, Air & Space Museum, the International Center of Photography, the Frick Collection and the Cooper Hewitt, were gone by Friday afternoon.

Tickets to the Brooklyn Museum, the Children’s Museum of the Arts, and other institutions were still available. And eight cultural institutions that were not previously involved with the program have inquired about participation, said Mr. Balber.