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 were also musicians.
He began playing the drums professionally as a teenager. He earned cash working in bars. When he was twenty he was touring with the Jess Stacy All-Stars featuring Lee Wiley. After the war he worked as the house drummer at Eddie Condon's in New York from 1947 until 1951.
Over the years in the 1950's and 1960's he worked in clubs in New York, Chicago and Boston and played with musicians such as Jimmy McPartland, Doc Cheatham, Vic Dickenson, PeeWee Russell, and Wingy Manone. 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 and The Dukes of Dixieland.
In 1973, after touring Europe and America, he and his brother Al, and nephew Sonny formed the Drootin Brothers Jazz 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 1980's.