Ida Kavafian was born in Turkey to Armenian parents. She moved with her family to America in 1956, and began studying violin in Detroit at age six. Her teachers included Ara Zerounian, Mischa Mischakoff, Oscar Shumsky, and Ivan Galamian, the last two of which she studied under while attending the Juilliard School from 1969 to 1975.
Her first major exposure came when she won the Vianna da Motta International Violin Competition in Lisbon in 1973. She won the Young Concert Artists International Auditions in 1978, which led to her New York recital debut at Carnegie Hall. That same year she became a member of the Tashi ensemble with Peter Serkin, who also accompanied her for her New York solo debut.
Kavafian began performing with her sister, Ani Kavafian, in 1983, when the pair played together at Carnegie Hall. In 1983 and 1984 she toured with Chick Corea. Kavafian has been a member of the Chamber Music Society from 1989–1993 and 1996–2002, she played with the Beaux Arts Trio from 1992 to 1998 and sporadically thereafter.
Kavafian founded her own group, Opus One in 1998, with Anne-Marie McDermott, Steven Tenenbom, and Peter Wiley. She teaches at the Curtis Institute of Music, the Juilliard School and the Bard College Conservatory of Music. She plays a J. B. Guadagnini violin made in Milan in 1751, and a Moes and Moes viola made in 1987.
Kavafian's repertore includes classical and early romantic works by Mozart, Beethoven, and Mendelssohn, as well as 20th century works of Ruth Crawford Seeger, Charles Wuorinen, and Toru Takemitsu. Takemitsu has composed a concerto for Kavafian.
Kavafian is also notably the founder of prestigious music festivals: the Bravo! Vail Valley Music Festival and the Music from Angel Fire.