USA Green Card- Child Born After the VISA?

I’m one of the 2010 DV lottery winners and had our interview on 14th November 2009 subsequently got our VISA on 20th December 2009 . My wife was expected on that time and delivered the baby on 5th January 2010 . The VISA officer mentioned that the baby doesn’t need to have a US VISA to travel since she was born after the issuance of VISA to us. The VISA office mentioned that we need to carry our baby’s long form birth certificate and a separate pport. My question is will Airline accept this situation since the baby hasn’t got any VISA? And what’s the procedure to apply for a SSN for the Baby since she hasn't got any VISA? Thank you