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. Alongside his classmates Orlando Cole, Max Aronoff, and Benjamin Sharlip, Brodsky formed in 1932 an ensemble which would later be called the Curtis String Quartet and served as the first violinist of the quartet until the group disbanded in 1981 after the death of the quartet's violist, Max Aronoff.
Brodsky joined the faculty at the Curtis Institute in 1932 and remained there until just after World War II when, with the rest of the Curtis String Quartet, he resigned over disagreements with certain of the school's policies to help found the New School of Music. After re-joining the faculty in the early 1950s, he remained for nearly fifty years, later being appointed to the Efrem Zimbalist Chair of Violin Studies, which position he held until his death in 1997. A respected pedagogue, his students are dispersed widely among the finest musical institutions in the world. Numbered among his students are Hilary Hahn, Joseph de Pasquale, Leila Josefowicz, Choong-Jin Chang, Juliette Kang, Judith Ingolfsson, Herbert Greenberg, Joey Corpus, 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.