Workers at the 67th Street Library on the Upper East Side of Manhattan turn away at least 10 people from every reading. They have been so overwhelmed by the rush at story time — held in the branch’s largest room, on the third floor — that once the space is full, they close the door and shut down the elevator. “It is so crowded and so popular, it’s insane,” Jacqueline Schector, a librarian, said.

Story time is drawing capacity crowds at public libraries across New York and across the country at a time when, more than ever, educators are emphasizing the importance of early literacy in preparing children for school and for developing critical thinking skills. The demand crosses economic lines, with parents at all income levels vying to get in.

Many libraries have refashioned the traditional readings to include enrichment activities such as counting numbers and naming colors, as well as music and dance. And many parents have made story time a fixture in their family routines alongside school pickups and playground outings — and, for those who employ nannies, a nonnegotiable requirement of the job.

In New York, demand for story time has surged across the city’s three library systems — the New York Public Library, the Brooklyn Public Library, and the Queens Library — and has posed logistical challenges for some branches, particularly those in small or cramped buildings. Citywide, story time attendance rose to 510,367 people in fiscal year 2015, up nearly 28 percent from 399,751 in fiscal 2013.