Born in Georgetown, Mississippi, he played for Vallejo High School in the old North Bay League after his family moved to Vallejo, California. Bass blossomed as a three-sport star at Vallejo High, where he ran for 3,690 yards and scored 68 touchdowns in 18 games. Bass scored a state-record 37 touchdowns in 1954, when he led the Apaches to an undefeated season at 9-0. The team averaged 54 points per game in 1954. On May 25, 2012, the Vallejo High School football practice field was officially dedicated as "Dick Bass Field".
Bass went on to star at College of the Pacific, now University of the Pacific, where Time Magazine called him a "One-Man Show" in 1958, after he ran for 700 yards in six games to become the season's leading NCAA ground gainer, while passing for the Tigers as well. He was a 1958 All American and was a Charter Member of the Pacific Athletics Hall of Fame in 1983. As a senior in 1958, Bass led the nation in rushing with 1,361 yards, including a dazzling display in the season opener in Berkeley where he gained 215 yards and scored one touchdown in the Tigers' win over a Cal team that would reach the 1959 Rose Bowl.
After being taken by the Rams as the second pick in the 1959 NFL Draft, he was named to the Pro Bowl three times, in 1962, 1963, and 1966. He rushed for 1,000 yards in a season 2 times (1962 and 1966). He finished his career with 5,417 yards rushing. He died at age 68 in Norwalk, California. Following his retirement, he worked as a color analyst on Rams radio broadcasts from 1977 to 1986.
In his autobiography, Los Angeles gang member "Monster" Kody Scott, alias Sanyika Shakur, reported that his mother identified Bass as his father.