Born in Kharkiv, in the Kharkov Governorate of the Russian Empire (in present-day Ukraine), he began his violin studies with his violinist father at the age of six. He later studied at the conservatory in Tbilisi, Georgia, and by 1926, was performing successfully all over the Soviet Union. That same year, he went to Paris to study with Lucien Capet. There he also played for Sergei Prokofiev and performed with pianist Vladimir Horowitz and violinists Nathan Milstein and Mischa Elman.
In 1930 he moved to America to study with Efrem Zimbalist at the Curtis Institute of Music, where he befriended Boris Goldovsky, a fellow Russian Jew. Brodsky and three other students formed an ensemble which would later be called the Curtis String Quartet and served as the first violinist of the quartet until group disbanded in 1981 after the death of the quartet's violist, Max Aronoff.
Brodsky was hired as a faculty member of the Curtis Institute in 1932, and later was appointed to the Efrem Zimbalist Chair of Violin Studies, which he held until his death in 1997. A respected pedagogue, his students are dispersed widely among the finest musical institutions in the world. Any given year there are about a dozen Brodsky students playing in the Philadelphia Orchestra. Numbered among his students are Jane E. Gilbert, Hilary Hahn, Joseph de Pasquale, Leila Josefowicz, Joey Corpus, Juliette Kang, Judith Ingolfsson, Herbert Greenberg, and Chin Kim.
With Aronoff, Brodsky founded the New School of Music in Philadelphia when they decided that there was a present need to train musicians specifically for a career in chamber music or in orchestra. In 1986, The New School of Music was merged into Temple University's Boyer College of Music and Dance, where Brodsky was appointed Professor Emeritus. He taught at the school until his retirement in 1996.
He died in Ocala, Florida.
- Goldovsky, Boris; and Cate, Curtis. My Road to Opera: The Recollections of Boris Goldovsky, p. 144. Houghton Mifflin, 1979. ISBN 0-395-27760-4. Accessed August 31, 2011. "Among the students at the Curtis Institute of Music I had made at least one friend. Of Russian Jewish origin like myself, he was named Jascha Brodsky and played the violin."