Costigan spent most of his youth in Colorado, where his parents moved in 1877. He graduated from Harvard University in 1899 and began his law practice in Denver the next year. His interest in good government led him to join nonpartisan organizations designed to improve municipal and state government in Colorado, including the Direct Primary League, which he helped organize. A Republican and a supporter of Theodore Roosevelt, Costigan bolted the party to help organize the Progressive Party of Colorado in 1912 and to become its unsuccessful candidate for governor in that year and again in 1914. As an attorney he represented the Denver Chamber of Commerce and the United Mine Workers of America in federal litigation.

In 1916 he became a supporter of President Woodrow Wilson, who later appointed him to the newly created Tariff Commission. He served with distinction on the commission as an advocate of a flexible tariff until he resigned in 1928 in protest against the high-tariff policies of the Calvin Coolidge administration. Costigan reentered Colorado politics as a Democrat and was elected U.S. senator in 1930. He proposed far-reaching legislation to meet problems created by the Great Depression and worked effectively for New Deal measures of President Franklin D. Roosevelt. He was cosponsor of the Jones-Costigan Act, a sugar-quota measure, as well as of the Costigan-Wagner Bill, an antilynching proposal that never became law. Ill from overwork, he gave up his Senate duties in 1936.