Finger was born was born on March 7, 1887 in Bielitz, Austria (now Bielsko, Cieszyn Silesia, Poland) to Henri and Hani Steifter Finger. After finishing high school and technical training, he moved to the United States in 1905.
Finger settled first in New Orleans then moved to Houston, Texas, three years after. He married Houston-native Gertrude Levy on June 18, 1913. He was a member of Congregation Beth Israel in Houston. He belonged to social organizations including: the Benevolent and Protective Order of Elks, the Houston Chamber of Commerce, Houston Turn-Verein, the Independent Order of B'nai Brith, and the Westwood Country Club. Finger was the father of Joseph Seifter Finger, a landscape architect and golf-course designer also locally active.
Starting in 1908, Finger performed architectural work in Houston, his first position at the Houston-branch office of Dallas architect C.D. Hill and Company. From 1912 to 1923, he worked as a partner with three different firms. He started his own practice in 1923, which he continued for about twenty-one years until he setup a partnership with George W. Rustay.
Finger designed the 1939 Houston City Hall, designed in a stripped classical style. In response to criticism from Houston mayor R. H. Fonville, who wanted a style with more classical reference, Finger said, "Here in America we are rapidly developing our own type of architecture which is far above that of foreign countries. We are building for the masses, not the classes."
Above the lobby entrance of the City Hall is a stone relief of two men taming a wild horse, symbolizing a community coming together to form a government to tame the world around them. This sculpture, and the twenty-seven other friezes around the building, were carved by Beaumont artist Herring Coe and co-designer Raoul Josset.
As the city's foremost Jewish architect in his time, Finger designed many Jewish landmarks, including the 1925 Congregation Beth Israel Temple, now the Heinen Theatre.
Finger's other work includes:
- Texas State Hotel (Now Club Quarters) Houston, 1929
- Montgomery County Courthouse, Conroe, Texas, 1935
- Heights State Bank (later became Rockefeller's, a major Houston music venue)
- Houston Municipal Airport Terminal, now the 1940 Air Terminal Museum
- Harris County Courthouse, Houston, 1953 (with partner Rustay, his last design)
- Barker Brothers Studio, Houston, 1931 now Lawndale Art Center
- James M. and Jessie West Mansion, Houston, 1929
- Fox, Stephen. "Finger, Joseph". Texas Handbook Online. Texas State Historical Association. Retrieved December 27, 2014. Uploaded on June 12, 2010.
- "Protected Landmark Designation Report: Sterne Building". City of Houston Archaeological and Historical Commission. March 21, 2011. Retrieved December 27, 2014.