There's a website called Ancestry24 that has records for South Africa.
According to the Birth Records page in their Learning Centre, birth registration was first introduced in South Africa in the late 1800s and wasn't mandatory in all regions until 1905. So whether you'll be able to find an 1884 birth record or not seems to be dependent on the location and when it began registering births. However, you may be able to find birth information in a baptism record, if that's applicable to your wife's family. A list of birth & baptism databases available at Ancestry24 can be found on the Birth Records page as well.
Further down the page, it also says that all birth records are indexed, and gives the archives which house the indexes.