Dague was born in Fowler, Indiana in 1885. His father, William Henry Dague, Sr., was a banker who was born in Pennsylvania in December 1844. At the time of the 1900 U.S. Census, Dague was living in Benton, Indiana with his father and two adult brothers. His mother had died. One brother, Maynard, was born in April 1879 and working as a miner. The other, Samuel, was born in August 1877 and was working as a lawyer.