New York State maintains an index of births (also deaths and marriages) on microfiche. Older records (75 years perhaps) are available for the public to browse. Not only don't you need the exact date of birth, you don't even need to know the exact year! I've spent a lot of time browsing these records, very profitably. I'm astonished that Pennsylvania doesn't do the same. Of course once you have the information you still have to order the certificate, but that's a much smaller problem.