Internet Archive has two digitized books, Births Reported in 1901, Borough of Manhattan; and Births Reported in 1902, Borough of Manhattan, both published by the New York State Health Department.
These books include a list of births, arranged alphabetically by last name (surname). Be sure to check the last pages for any additional names, not in sequence.
What Can I Learn from Manhattan, New York City Birth Records?
These New York City birth records include:
Child’s First Name, Last Name, and Middle Initial (if there is one)
Child’s Date of Birth
Birth Certificate number