Births weren't always recorded that far back. There is an online birth index for Orleans Parish on the Orleans Parish GenWeb archives page, but she's not listed. If she's not there, I'd say no official birth record exists. (
http://usgwarchives.org/la/orleans.htm)
As for a baptism record, it would generally be filed with the individual church. Would you happen to know which church the family attended? If she was Catholic, it's possible the record might be with the Archdiocese, but I'm not familiar with researching Catholics.