Binney was born in Philadelphia, Pennsylvania, the son of Dr. Barnabas Binney (1751–1787), a prominent Philadelphia physician who cared for Deborah Sampson.
 He graduated from Harvard College in 1797, where he founded the Hasty Pudding Club in 1795.
	
Through his sister Susan Binney Wallace, he was the uncle of Horace Binney Wallace (1817–1852), a legal critic and through his sister, Mary Sarah Binney Sargent (d. 1824), wife of Lucius Manlius Sargent (1786–1867), an author and temperance advocate, he was the uncle of well-known author and Horace Binney Sargent (1821–1908), a Civil war veteran.
Binney was born in Philadelphia, Pennsylvania, the son of Dr. Barnabas Binney (1751–1787), a prominent Philadelphia physician who cared for Deborah Sampson.  He graduated from Harvard College in 1797, where he founded the Hasty Pudding Club in 1795. 	
Through his sister Susan Binney Wallace, he was the uncle of Horace Binney Wallace (1817–1852), a legal critic and through his sister, Mary Sarah Binney Sargent (d. 1824), wife of Lucius Manlius Sargent (1786–1867), an author and temperance advocate, he was the uncle of well-known author and Horace Binney Sargent (1821–1908), a Civil war veteran.
Horace Binney