Born in Beemerville, Wantage Township, New Jersey, Stivers attended common and private schools and Mount Retirement Seminary in Wantage.
He taught school. He engaged in mercantile pursuits in Ridgebury and later in Middletown from 1855 to 1864. He served as clerk of Orange County 1864-1867 and resided in Goshen, New New York He returned to Middletown and became proprietor of the Orange County Press in 1868 and was also one of the proprietors and editors of the Middletown Daily Press.
He was appointed by President Grant as United States collector of internal revenue for the eleventh district of New York in 1869 and served until 1883.
He served as delegate to the Republican National Convention in 1880. He engaged in banking.
He was an unsuccessful Republican candidate for election in 1884 to the Forty-ninth Congress to fill the vacancy caused by the death of Lewis Beach and for election in 1886 to the Fiftieth Congress. Stivers was elected as a Republican to the Fifty-first Congress (March 4, 1889 – March 3, 1891).
He was not a candidate for renomination in 1890.
He engaged in banking. He died in Middletown, New York, February 2, 1895. He was interred in Hillside Cemetery.