Hi,
In the past, usually the Godmother held the child. But more recently it has become the custom that the parent should be the one since the parents have co-operated with God in the creation of the child and also because they have the first obligation to educate their children about God. Even a non-Catholic parent ought to hold the child. If for some reason, the parent cannot, then the Godparent ought to.
Fr. Vincent Serpa, O.P.