Answer:
Basically you cannot find a Person's Birth Date Absolutely Free, as there is your time, access to the Internet, unless you have Free Access, and what I am thinking you are wanting to know is there a Web Site that is Free that you can possibly find out a person's Birth Date and yes there is one site FamilySearch.org
Then the question becomes also are you Asking Living or someone Deceased, When were they born about, where born and if this record is available. As not every living person maybe recorded & available to find, same of the deceased, without more than a name, about when born and where.
And why there are those as myself, who are Genealogy Researchers, also the intent & purpose of this information to be known or reason.
And there are Websites that are Free to Search, however cost to view full results and knowing which one if more than one result, is the person you are wanting to know this information.
Another avenue to search is Rootsweb.com http://www.rootsweb.ancestry.com/
Archives - http://archiver.rootsweb.ancestry.com/cgi-bin/search
or Join for Free one of Rootsweb's Genealogy Surname Mailing List http://lists.rootsweb.ancestry.com/index/
Another Site is GenForum - http://genforum.genealogy.com/
Or find a Volunteer Genealogy Researcher who may search for Free. There are no Guarantees, even if you use an Online Search Engine of Choice, you can find the Person in which you seek this information, is in Fact the Same Person.
Collaborating Information if Living or Deceased, is Needed in most cases, Full Name, where Last Living at time of Birth or Death.
D.R. "Doc" Young - drdocyoung.com