If God was disgusted with us after Adam and Eve, why would He allow babies to be born?

God's anger had nothing to do with this. The actions of Adam and Eve showed that they had become disobedient, and had damaged their spiritual relationship with God. Rather than destroy them, God immediately had plans for a Messiah to come and restore mankind to him. That is why God said to the woman, thy seed shall crush the head of the serpent. Always remember that along with justice and power, wisdom and truth, God is also love, mercy and grace.