Thanks to the German Genealogy Group and the Italian Genealogical Group, there is now an online index to New York City births; 1901-1907 are available now, and the complete index will run from 1881-1909.
I've checked out my families of interest and realize I will have some work to do to sort out the FINKs and BANKs who were in New York that early.
A caveat: researchers should know that about a quarter of all pre-1910 births were not reported as many took place at home and attending doctors or midwives did not report the event to the authorities. If your ancestors are not in this index, go to census records and other supporting documents.
Volunteers, representing many ethnic and religious groups, have spent many hours compiling these indexes and others already completed (other vital and naturalization records). The amazing John Martino of the IGG is the spearhead of this volunteer army.
For more information, visit the German Genealogy Group or the Italian Genealogical Group.
Additionally, Steve Morse has already created a One-Step search tool for these records.
The birth records are on microfilm at the NYC Municipal Archives; copies made be ordered online or by mail. For more details, click here.
The microfilms of these records are always available at the main Family History Library in Salt Lake City. Researchers elsewhere in the US and many other countries (not in Israel) may order the films from SLC and view them at their local Family History Center branches. Search the FamilySearch.org catalog for each borough of New York City to find the microfilm reel numbers.