Born in Dumfriesshire, Scotland, Beck immigrated to the United States in 1838 and settled in Wyoming County, New York. He moved to Lexington, Kentucky in 1843 and graduated from Transylvania University in 1846. Beck was admitted to the bar and commenced the practice of law in Lexington. Until shortly before the Civil War, he was law partner of John C. Breckinridge, the U.S. Vice President who became a Confederate general; during the Civil War, Beck was interrogated by a military commission about his knowledge of his former partner's activities.
After the war Beck was elected as a Democrat to the United States House of Representatives serving Kentucky's district 7. He was appointed to the Committee on Reconstruction where it was expected that as a newcomer and an immigrant he would be no obstacle to Republican intentions, but he immediately became a tenacious advocate of the rights of the defeated states. He was elected to the Fortieth and to the three succeeding Congresses, serving in all from March 4, 1867 to March 3, 1875.
In 1876, Beck was appointed a member of the commission to define the boundary line between Maryland and Virginia. He was then elected to the United States Senate in 1876, being reelected twice and serving in all from March 4, 1877, until his death in Washington, D.C., on May 3, 1890. While in the Senate, Beck was the Democratic Conference Chairman from 1885 to 1890, and the chairman of the Committee on Transportation Routes to the Seaboard. He was prominent in the discussion of tariff and currency questions.