Drootin was born in Kiev, Ukraine and moved to Boston, Massachusetts with his family when he was five. His family was musical going back many generations. His father played the clarinet and two of his brothers as well as his nephew were also musicians.
He began playing the drums professionally as a teenager. When he was twenty he was touring with the Jess Stacy All-Stars featuring Lee Wiley. He also toured with Ina Ray Hutton around that time. After those tours he did a stint with Wingy Manone in Chicago and on the road. After the war he worked as the house drummer at Eddie Condon's in New York from 1947 until 1951.
In the 1950s and 1960s he worked in clubs in New York, Chicago and Boston. He did a stint as bandleader at New York's El Morocco Club. He and his brother Al were in the house band at George Wein's Storyville in Boston during the early and mid 50's. In those years he played with musicians such as Bobby Hackett, Jimmy McPartland, Doc Cheatham, Vic Dickenson, Claude Hopkins, Arvell Shaw and Pee Wee Russell.
He also recorded with Tommy Dorsey, Bobby Hackett, Jack Teagarden, Eddie Condon, Ruby Braff, Anita O'Day, George Wein, The Newport All-Stars, Lee Konitz, Sidney Bechet, PeeWee Russell and The Dukes of Dixieland. In 1968/69 he toured and recorded with Wild Bill Davison's Jazz Giants and then formed "Buzzy's Jazz Family" borrowing some of Wild Bill's sidemen (Herb Hall, Benny Morton) and adding Herman Autrey on trumpet and his nephew Sonny Drootin on piano.
In 1973, after touring Europe and America, he returned to his hometown of Boston where he and his brother Al (sax and clarinet), and nephew Sonny formed the Drootin Brothers Band. They played at the Newport Jazz Festival. Buzzy played at the very first Newport festival and at many of the festivals after that. He also played at the Los Angeles Classic Jazz Festival in the 1980s.