Aiken was born in Dummerston in Windham County, Vermont, and graduated from Brattleboro High School while living in Putney, Vermont in 1909. A Republican, he was elected to the Vermont House of Representatives in 1931 and served as Speaker of the House from 1933 to 1935. He was lieutenant governor of Vermont from 1935 to 1937 and subsequently served two terms as governor, being first elected in 1936 and re-elected in 1938.

As governor of Vermont, he broke the monopolies of many major industries, including banks, railroads, marble companies, and granite companies. He also encouraged suffering farmers in rural Vermont to form co-ops to market their crops and get access to electricity.

He was elected to the United States Senate on November 5, 1940, to fill the vacancy in the term ending January 3, 1945, caused by the death of Ernest W. Gibson, and was re-elected in 1944, 1950, 1956, 1962, and 1968. During his time in the Senate he served in a number of leadership roles including Chairman of the Committee on Expenditures in Executive Departments in the 80th Congress and in the Committee on Agriculture and Forestry in the 83rd Congress. He was one of the white-haired men during the time of President John Fitzgerald Kennedy's inaugural statement about the torch passing to a new generation. During the Vietnam war, he said the US should declare victory and bring the troops home, which turned out to be a very perceptive strategy that was adopted too late.

He was a proponent of many progressive programs such as Food Stamps and public works projects for rural America, such as rural electrification, flood control and crop insurance. His views were at odds with those of many Old Guard Republicans in the Senate. He was often called a communist. Vermonters, however, gave him such respect that he reportedly spent only $17.09 on his last reelection bid.

He married Beatrice Howard and had four children by her: Dorothy, Marjorie, Howard, and Barbara. He married his second wife, Lola Pierotti, on June 30, 1967.