Stránský grew up in Prague and studied medicine, taking parallel music lessons with Salomon Jadassohn in Leipzig and Robert Fuchs in Vienna. In 1896, Josef Stránský returned to Bohemia and took the state medical exams before deciding to pursue a musical career. In 1898, Stránský got his first appointment at the German Theater in Prague. He served as the bandmaster at Hamburg City Theater between 1903 and 1909, later taking over as the conductor and concertmaster of Symphonisches Blüthner-Orchester Berlin for one season.

In 1911, Josef Stránský relocated to the United States and became the principal conductor of the renowned [url=https://discogs.com/artist/388185]New York Philharmonic[/url]. Taking over legendary Gustav Mahler, Stransky was only thirty-nine (the youngest NY Phil conductor in 50+ years, since Carl Bergmann (1821—1876) took the same role in 1855) and practically unknown to the US audience, so he faced unprecedented criticism in the press. Musical America compared Stránský's appointment with "Aesop's fable of the mountain in labor which finally brought forth a mouse," and The New York Times implied that the [url=https://discogs.com/label/853957]Philharmonic Society[/url] selected Josef due to lower financial demands versus other candidates like Oskar Fried or Bruno Walter. At the same time, H. L. Mencken's American Mercury magazine claimed the controversial appointment was driven "not so much by his musical abilities as by his social charm and personal cleverness;" composer and music critic Daniel Gregory Mason called the conductor "a total music incompetent."

Despite the severe initial backlash, Josef Stránský had a prolific and fruitful tenure with the New York Philharmonic, expanding its repertoire wider than any previous conductor. During the First World War, he favored English, Russian, and French composers in the Orchestra's programming. Stránský also premiered numerous contemporary American composers, such as George Whitefield Chadwick, Arthur Foote, Edward MacDowell, John Knowles Paine, and John Philip Sousa. He conducted the earliest New York Philharmonic recordings for the Columbia Graphophone Company in 1917. After the ensemble merged with the National Symphony Orchestra, Josef Stransky shared his final 1922/23 season with a newly appointed Willem Mengelberg.