He was born in Little Britain in 1735, the son of Irish immigrants/refugees John and Mary Crawford Young who came to America during the late 1720s. During the 1760s, Joseph Young accompanied his older brother to Albany and later took over Thomas Young's medical practice. In 1761, he married Sharon Brown of Amenia, Dutchess County - where he lived prior to coming to Albany.
By 1766, he was identified as an Albany home owner. Earlier that year, he joined his brother in signing the constitution of the Albany Sons of Liberty. His home was located on Albany's Southside near the Cartwright's King's Arms Tavern.
In 1775, he joined the crusade for American liberties when he was elected to the Albany Committee of Correspondence. He served until the end of 1776. At the same time, he had been appointed surgeon to the Ninth Connecticut Regiment of the Continental line. He later received a land bounty right in conjunction with the First Regiment of the Albany County militia.
Following the death of his brother in 1777, Thomas Young's widow and son moved to Albany and in with Joseph.
In 1778, he was made director of the Albany hospital. In 1781, he was promoted to hospital physician surgeon and served to the end of the war.
He was the author of one of the first if not the first medical textbook in United States
After the war, Joseph Young removed to New York City where he continued to practice medicine until his death in 1814.
He trained a number of important physicians including the artist-engraver Alexander Anderson.