from their website

"Effective November 19, 2007, individuals who are renewing a current Ohio driver license, Ohio commercial driver license (CDL), or Ohio identification (ID) card will no longer be required to present a Social Security card at an Ohio Bureau of Motor Vehicles (BMV) license agency for verification at the time of renewal.

An Ohio driver license, Ohio CDL, or Ohio ID card, which is current, or has been expired for less than six months, is acceptable without further proof of the applicant’s Social Security number if it has been previously verified in the BMV’s records. "

http://bmv.ohio.gov/driver_license/aid.htm
for further info