If your child was born in the USA, he or she is a US citizen (even if you are not).
Your child is entitled to a US passport. Contact your local authorities and get a copy of your child’s birth certificate. Certain Post Offices have been appointed as passport agencies. (Be aware that some of these Post Offices require an appointment, even if their page on the USPS website says they don’t require it. Call to be safe!) You can get the application form at the Post Office and also have the pictures taken there.
Because you are a South African citizen, your child is also entitled to South African citizenship. Here is how to register your child’s birth in South Africa.