He was raised in Providence, Rhode Island and attended the public schools. He graduated from Hope Street High School in Providence in 1902, from Yale University in 1906, and from Harvard Law School in 1909.
In 1911, he was admitted to the bar and began the practice of law in New York City. He returned to Providence in 1913, and continued practicing law. Aldrich was elected as a Republican candidate to the Sixty-eighth Congress and to the four succeeding Congresses, serving from March 4, 1923 to March 3, 1933.
He was not a candidate for renomination in 1932.
While in Congress, he spoke out against the Immigration Acting of 1924, that limited the annual number of immigrants who could be admitted from any country. After leaving Congress, he resumed his legal career in Providence until his death there on December 25, 1941.
He is interred in Swan Point Cemetery in Providence. Aldrich was born into a family descended from John Winthrop, William Wickenden, Roger Williams and John Steere.
He was the son of Nelson West. Aldrich and Abby Pearce Chapman.
His father was a leader of the Republican Party in the Senate, where he served from 1881 to 1911.
He was a Republican member of the United States. House of Representatives, and served in the Rhode Island State Senate and the Rhode Island House of Representatives. He moved to Warwick, Rhode Island and became involved in politics and was a member of the Rhode Island House of Representatives from 1914-1916, and served in the Rhode Island Senate from 1916-1918. In July 1923 he became a member of the Rhode Island Society of the Sons of the American Revolution.