Their children said Tuesday that the couple died "peacefully and under circumstances of their own choosing" at a Zurich clinic run by the group Dignitas.
"After 54 happy years together, they decided to end their own lives rather than continue to struggle with serious health problems," said a statement from the couple's son and daughter, Caractacus and Boudicca.
The statement said Downes, who became Sir Edward when he was knighted by Queen Elizabeth II in 1991, had become almost blind and increasingly deaf. His 74-year-old wife, Joan, was a former dancer, choreographer and television producer who had devoted years to working as his assistant. British newspapers reported that she had been diagnosed with cancer.
The deaths are the latest in a series of high-profile cases that have spurred calls for a legal change in Britain, where assisted suicide and euthanasia are banned.
Born in 1924 in Birmingham, England, Edward Downes studied at Birmingham University, the Royal College of Music and under German conductor Hermann Scherchen.
In 1952, he joined London's Royal Opera House as a junior staffer -- his first job was prompting soprano Maria Callas. He made his debut as a conductor with the company the next year and went on to become associate music director. Throughout his life, he retained close ties to the Royal Opera, conducting 49 different operas there over more than 50 years.
He also had a decades-long association with the BBC Philharmonic Orchestra, where he became principal conductor and later conductor emeritus.
Downes was known for his support for British composers and his passion for Prokofiev and Verdi, on whom he was considered an expert.
In the 1970s, he became music director of the Australian Opera, conducting the first performance at the Sydney Opera House in 1973. He also worked with the Netherlands Radio Orchestra and ensembles around the world.
The couple are survived by their children, who said their parents "both lived life to the full and considered themselves to be extremely lucky to have lived such rewarding lives, both professionally and personally."