Babcock was born in Westerly in 1707 to James Babcock and Elizabeth Saunders, who were amongst Westerly's earliest settlers. Babcock became a Seventh Day Baptist as a young man. In 1724 he graduated from Yale College as Yale's first Rhode Island graduate. He then studied medicine in Boston and London and returned to Westerly to practice medicine in 1734. Around this time he married and purchased the Babcock-Smith House. Babcock later served as a justice on the Rhode Island Supreme Court, serving as an associate justice from 1747 to 1748 and chief justice from 1749 to 1751 and 1763 to 1764.