I’ve already asked about the oldest free public library: now what’s the oldest children’s library in America?
The oldest children's library was probably a school library. According to the American Library Assocation (ALA), it is not entirely clear when the first school library was established:
The birth of America’s school libraries cannot be assigned a definite date. Rather, these first school libraries were born unheralded in the earliest colonial times when the teacher in the one-room placed the Bible, a chapbook, and the Bay Psalm Book on the corner of his desk.
The above is quoted from the article on "School Libraries" in the Encyclopedia on Library and Information Science. ALA goes on to say (emphasis mine),
The same article also reports that in 1740 Benjamin Franklin recommended school libraries as a key element in the ideal academy, and the Penn Charter School in Philadelphia designating a specially designed room as the library in 1744.
This might have been the oldest proper school library in America, but the article is not explicit about this.
The Association for Library Service to Children (ALSC) is a much more recent organisation. It's oldest precursor, the Section for Library Work with Children, had its first meeting in 1901. According to the ALA Handbook 1948 (with 1949 copyright), cited on the webpages History of the ALSC,
The Division of Libraries for Children and Young People was organized and adopted its constitution and by-laws in June 1941. On June 24, 1941, the division was approved by A.L.A. Council. The division is made up of the American Association of School Librarians (formerly the School Libraries Section which had its first meeting in 1915), the Children's Library Association (formerly the Section for Library Work with Children, which had its first meeting in 1901) and the Young People's Reading Round Table which first met in 1930.
ALA's Young Adult Services Division was established in 1957, but the article does not say anything about the history of libraries for teens.