Born in St. Joseph, Missouri, Welch was educated in the public schools.
He served as assistant director of Reconstruction Finance Corporation, Kansas City, Missouri, in 1946 and 1947.
He was not a candidate for renomination in 1952 to the Eighty-third Congress but was unsuccessful for the Democratic gubernatorial nomination.
He was a resident of St. Joseph, Missouri, until his death in Methodist Hospital April 26, 1963.
St. Joseph's baseball park, Phil Welch Stadium, is named in his honor.