By far the largest number of children's books—especially those for circulation (lending) to children and their families—is to be found at The New York Public Library. The largest collections of children's books in that you can visit are at its Children's Center at 42nd Street as well as the many children's rooms in the 87 neighborhood branches in Manhattan, the Bronx and Staten Island. There are also children's collections at the Brooklyn Public Library and the Queens Library.
However, there are also a number of smaller, often non-profit, children's libraries in New York City that are in most cases free and target the needs of children and adolescents with interests in science, art, literature, poetry and New York and American history—not to mention other libraries for children and families who need information on health issues, ethnic or religious concerns as well as organizations that provide lists and bibliographies of children's books to teachers, parents and librarians.
Central Park by peterjr1961, on Flickr
The New York Hall of Science in Queens has a science library that addresses the needs of pre-schoolers, elementary school and high school students. Admission to the museum and library is sometimes free.
The American Museum of Natural History Library is non-lending research library whose resources are primarily devoted to advanced study of the geological and biological sciences. However, its holdings are open to advanced secondary school students—without admission to the museum—who can make use of its resources onsite. Appointments: 212.313.7593.
The Metropolitan Museum of Art has extensive resources for children including the Nolen Children's library (to which admission is free at all times) that is focused on art education for children. Their webpage "For Kids" links to a variety of activities, projects and games at a number of other children's art libraries and collections.
"The Uni Project" seeks in New York City and elsewhere to create a "portable reading room" that is "is dedicated to expanding a culture of learning beyond the walls of schools and libraries and into public space" where it "shares books, showcases the act of learning and improves public space"—frequently in public parks—to children in Manhattan, Brooklyn and Queens.
Reading, via Wikimedia Commons
The Center for Fiction in Manhattan has a number of free programs and events including readings by authors for children and young adults that encourage New York City children to read.
The New York Society Library, in Manhattan, has a circulating children's collection that is open to the children of its members who pay a fee. It does have readings by children's authors that are often free and open to the public.
Poet's House in lower Manhattan has a "Children's Room" with many poetry books that appeal to children and it also has a number of programs that it conducts and outreach to encourage children's interest in poetry.
New York City and United States History: The New-York Historical Society ("NYHS") is located at 170 Central Park West [at West 77th Street.] Within the NYHS is the Dimenna Children's History Museum that lets children ages 8-13 "explore New York through the eyes of children in the past" as well as the Barbara K. Lipman Children's History Library that holds: story books, history books and games that concern New York and American history. Admission is charged.
Ethnic and Religious Materials
The Jewish Children's Museum in Brooklyn, N.Y. also offers a children's library with a focus on Jewish cultural and religious issues. And there are numerous children's libraries and book services that serve New York City's ethnic and religious groups, for example, the free online and multi-lingual International Children's Digital Library; PJ Library (that mails Jewish children's literature in the New York City area) and Pauline Kids (Roman Catholic books).
Consumer Health Issues
The NYU Langone Patient and Family Resource Libraries has three locations in Manhattan that provide free consumer health information of interest to children, adolescents and their parents and takes questions by email, telephone and US Mail. The New York City Department of Health and Mental Hygiene provides definitions and discussion of health conditions with: "Health Topics A-Z" It also maintains the William Hallock Park Memorial Public Health Library open to residents by appointment at 347.396.2915. Planned Parenthood of New York provides information to adolescents and adults on sexual and reproductive health issues as well as its online "Topics." Visually Impaired High School Students and Adults who wish to attend college may obtain books and academic assistance at the Jewish Guild for Healthcare.
Resources: Those Who Teach, Write or Curate Books for Children
The American Library Association's Association for Library Service to Children provides current and past annual lists of books selected by librarians as Children's Notable Books as well as many other works of children's literature that have received Awards and Recommendations and "Great Websites for Kids."
The Bank Street College of Education Library provides free online "book lists for children" on all sorts of topics for children and young adults including: memoirs, poetry, short stories, dialect variation and LGBT issues.
The Children's Books Council in Manhattan is the publisher's trade organization for children's books that compiles annual reading lists of Children's Choices, Teacher's Choices and Young Adult Choices.