Curran began his law practice with the New York State Insurance Fund, followed by three years in private practice. In 1928, he became an Assistant United States Attorney in the Southern District of New York. He left federal service in 1931 to run as a Republican for the New York City Board of Aldermen. He was unsuccessful in his first campaign, but was elected an alderman two years later and was chosen Minority Leader in 1934. In 1936, he was elected as Republican leader of his district, and in 1937 was elected as a delegate to the New York State Constitutional Convention of 1938. He became leader of the New York County Republican organization in 1940 and served in that post until his death. He ran unsuccessfully for the United States Senate in 1944, losing the election to incumbent Robert F. Wagner.
Curran served as Secretary of State of New York from 1943 to 1955, appointed by Governor Thomas E. Dewey. As host of the state's 1944 meeting of presidential electors, he supported the elimination of the Electoral College. In his speech before the Democratic electors, he asserted that this would likely be the last such gathering, since so many Americans regarded the procedure as outmoded.
