Baker was born in Lipton on November 24, 1915, and educated in Lipton and Regina. He enrolled in the Regina Teachers' College and taught from 1934 until the outbreak of World War II. He served two years in the RCAF training Commonwealth aircrews. Baker completed a business course in Chicago and was appointed secretary of the Saskatchewan Public Service Commission in 1944.
Baker was elected to Regina City Council in 1955. He was elected mayor three years later and re-elected the next three terms until Harry Walker defeated him in 1970. Baker was elected as a CCF member of the Legislative Assembly in 1964 and re-elected three times as an NDP MLA. In 1973 Baker was elected mayor again and re-elected in 1976, the longest serving mayor of Regina.
During his tenure as mayor, Regina undertook several major projects such as the construction of the Ring Road, the Centre of the Arts and the new City Hall. He was criticized for serving both as mayor and an MLA at the same time. Baker ran in the 1982 election but was defeated, ending his twenty-seven year career representing Regina. In 1997, the city of Regina re-named the city council chamber in his honour. Henry Baker died in Regina on March 4, 2004.