They definitely have fingerprints at birth, but since they're so tiny newborns are usually footprinted if prints are used for ID. And those footprints can identify the grown adult, so I assume the hands are the same.
I've heard that babies have fingerprints by 22 weeks gestation, and while the fingerprints stretch and grow, they stay the same basic shape. You can identify an adult from a fingerprint taken at 22 weeks gestation. But you'd need a really big magnifying glass.
All my babies had their foots and hands inked up and put on their hospital "birth certificate" (not the official ones, just a pretty one the hospital gave us). They have distinct fingerprint ridges at birth.