Do I need to enter an adult fingerprint to help my child apply for a social security card?

No need. According to Baidu's legal information, to apply for a social security card for a child, you need to bring your household registration book (copy of the first page and the child's household registration page), your parents' ID card (double-sided copy) and an inch photo of your newborn on a blue or red background (required in some places), and go to the street or community labor security office where your household registration is located to apply for insurance, without entering adult fingerprints.