Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
There is no particular Du’a that is said when a baby is born. However, the person should thank Allah, and express his gratitude to Allah on giving him/her a child, which is a great blessing of Allah. The right of Allah for this great blessing is your appreciation and expression of great gratitude to Him. Allah Says (interpretation of meaning): {And (remember) when your Lord proclaimed: "If you give thanks (by accepting Faith and worshipping none but Allâh), I will give you more (of My Blessings),…}[14:7].
One should always supplicate Allah to grant good guidance, steadfastness, bounty and blessing for his children. Allah Says (interpretation of meaning): {….and make my off-spring good….}[46:15]. Allah also Says (interpretation of meaning): {“O my Lord! Make me one who performs As-Salât (Iqâmat-as-Salât), and (also) from my offspring,….}[46:15]. Allah reported from Ibrahim that he used to make this Du’a: {“O my Lord! Make me one who performs As-Salât (Iqâmat-as-Salât), and (also) from my offspring, our Lord! And accept my invocation.}[14:40].
Allah knows best.