Born in Crailsheim, Baden-Württemberg, Meyer began playing the clarinet at an early age. Her first teacher was her father, also a clarinetist. She studied with Otto Hermann in Stuttgart and then with Hans Deinzer at the Hochschule für Musik und Theater Hannover, along with her brother, clarinetist Wolfgang Meyer, and husband, clarinetist Reiner Wehle, who played later in the Munich Philharmonic. She began her career as a member of the Bavarian Radio Symphony Orchestra and the Berlin Philharmonic, where her appointment as one of the orchestra’s first female members caused controversy. Herbert von Karajan, the orchestra’s music director, hired Meyer in September 1982, but the players voted against her at the conclusion of her probation period by a vote of 73 to 4. The orchestra insisted the reason was that her tone did not blend with the other members of the section, but other observers, including Karajan, believed that the true reason was her gender. In 1983, after nine months, Meyer left the orchestra to become a full-time solo clarinetist.

Sabine Meyer plays the clarinet and basset clarinet in B♭ and A, as well as a basset horn in F, all made of grenadilla by Herbert Wurlitzer, and clarinets in B♭ and in A made of boxwood, manufactured by Schwenk & Seggelke, which she mainly uses in chamber music. In 1984, Meyer had commissioned Wurlitzer to build a basset clarinet (in A) for her, not a historical replica, but a modern hitherto only occasionally built instrument. Since then, she has been playing the clarinet concerto by Mozart (and his clarinet quintet) in a reconstructed version.

By the 1990s, Meyer had become a prominent solo clarinetist, recording regularly and exclusively for the EMI label. These EMI recordings include a CD of French music for Clarinet and Piano with Oleg Maisenberg, entitled French Recital. A disc of clarinet concertos by Ludwig Spohr and Franz Krommer was released in July 2007, for which she collaborated with her student Julian Bliss.