In an answer to a question about New York records, efgen said:
The NYC birth index is actually available through 1965 on microfilm. However, only the indexes through 1909 are currently available online.
The Family History Library has a complete copy of the microfilmed indexes and records that are available at the NYC Municipal Archives. So once you have a certificate number from the index, you can order the appropriate microfilm through your local FHC to view the actual record.
Of course the Municipal archive only holds births through 1909, so it's unclear to me what records FamilySearch has.
Browsing the FamilySearch site, I was unable to find the metadata about the microfilm for the NYC births through 1965 mentioned by efgen. Their New York births collection covers the state, not NYC. Since I know the birth dates of the people from family records, I was thinking it should be possible to bypass the Municipal archive. Is this possible?