"Born at" is used in the field of genealogy. For example,
.. Name: John Smith. Born at Chicago, Cook, IL, 1909. Died at Buffalo, NY, 1945.
Name: Jane Smith. Born on 28 Jun 1806 at Oxford, CT. Died at Oxford, CT 1 May 1861.
'Born at" is also used in biographical writing, including entries in encyclopedias. For example:
John Smith, born at Chicago, Ill., on 11 November 1923, enlisted in the Navy 16 June 1941; was transferred to the Marine Corps at his own request
Mary Smith was born at Boston, MA in 1944. She studied at the University of Massachusetts and the University of California at Los Angeles.
These are both specialized usages. As PromegaX says, the usual preposition for someone's birthplace is in.