Drake, the eldest child of two American expatriates, was born in Paris, France. Her grandfather, Tracy Drake, and his brother had opened the Drake Hotel in Chicago on New Years Eve in 1920. The Drakes lost their money in the 1929 stock-market crash. As a result, she returned to the U.S. on the SS Île de France with her parents, brothers, and a nanny. She grew up in Chicago, Westport, Connecticut, Washington, DC, Virginia, North Carolina, and New York City. She went to 12 different schools, both private and public, before concentrating on theatre and acting at National Park Seminary.

Cary Grant first spotted her in 1947 while she was performing in London. The two, who both happened to be returning to the U.S. on the RMS Queen Mary, struck up an instant rapport. At the insistence of Grant, Drake was subsequently signed to a film contract by RKO Pictures and David Selznick, where she appeared, opposite Grant, in her first film, the romantic comedy Every Girl Should Be Married (1948). New York Times film critic Bosley Crowther called her performance “foxily amusing”.

On Christmas Day 1949, Drake and Grant married in a private ceremony organized by Grant's best man, Howard Hughes, and deliberately chose a low-key, introspective private life. They delved into transcendentalism, mysticism, and yoga. She took up causes including the plight of homeless children in Los Angeles. In 1954, they bought the "Las Palomas" estate in the Movie Colony neighborhood of Palm Springs, California.

The couple co-starred in the radio series Mr. and Mrs. Blanding (1951). They appeared together in the comedy-drama Room for One More (1952), and Drake appeared in a number of leading roles in England and the U.S., and a supporting role in the satiric comedy film Will Success Spoil Rock Hunter? (1957). Drake subsequently gave up acting to focus on her other interests, such as writing. Under the name Betsy Drake Grant, her novel Children, You Are Very Little (1971) was published by Atheneum Books. She worked as a volunteer and studied at the UCLA Neuropsychiatric Institute, and earned a Master of Education degree from Harvard University.

In July 1956, Drake survived the sinking of the Italian ocean liner SS Andrea Doria. At the time, she had been visiting Grant in Spain and was returning to the United States. She boarded the Doria, along with dozens of other wealthy travelers and tourists, at Gibraltar, which was one of many stops the ship made between her home port of Genoa and her final destination of New York. Drake sailed as a first-class Passenger, occupying a single cabin on the ship's boat deck. When the Doria collided with the Stockholm, Drake waited with the other passengers for rescue, as the ship's severe list rendered half the Doria's lifeboats useless. She was among more than 700 people rescued from the Doria by the famed French Passenger liner Ile de France.

Grant and Drake separated in 1958, remaining friends, and divorced in 1962. Their marriage constituted his longest union. Grant credited her with broadening his interests beyond his career and with introducing him to the then-legal LSD therapy, which he claimed helped him finally to achieve a degree of mental peace, and to hypnosis, which he used to successfully quit smoking. Later, Drake took LSD as a way of recovering from the trauma of divorce. Drake had no children with Grant.

Drake's last screen appearance was in the documentary film Cary Grant: A Class Apart (2005), in which she reflected on Grant and their time together, and denied rumors alleging he was bisexual.

Drake spent the latter part of her life in London, where she died aged 92 on October 27, 2015.