Robinson was born in St. Louis, Missouri, the daughter of Ralph Bowers Bowman and Helen (Hall) Bowman. Her older sister is the mathematical popularizer and biographer Constance Reid. The family moved to Arizona and then to San Diego when the girls were a few years old.
She entered San Diego State University in 1936 and transferred as a senior to University of California, Berkeley in 1939. She received her AB degree in 1940 and continued in graduate studies. She married Berkeley professor Raphael Robinson in 1941. She received the Ph.D. degree in 1948 under Alfred Tarski with a disseration on "Definability and Decision Problems in Arithmetic".
Her heart had been damaged by rheumatic fever as a child, and as an adult she suffered poor health and shortness of breath. In 1961 she underwent an operation to remove the scar tissue from her mitral valve. The operation was a success and she became much more active physically and took up bicycling for exercise.
In 1975 she became a full professor at Berkeley, teaching quarter-time because she still did not feel strong enough for a full-time job.
Hilbert's tenth problem asks for an algorithm to determine whether a Diophantine equation has any solutions in integers. A series of results developed in the 1940s through 1970 by Robinson, Martin Davis, Hilary Putnam, and Yuri Matiyasevich resolved this problem in the negative; that is, they showed that no such algorithm can exist.
George Csicsery produced and directed a one-hour documentary about Robinson titled Julia Robinson and Hilbert's Tenth Problem, that premiered at the Joint Mathematics Meeting in San Diego on January 7, 2008. Notices of the American Mathematical Society printed a film review and an interview with the director.
Robinson was attracted to politics by the 1952 presidential campaign of Adlai Stevenson. (Stevenson was her husband's first cousin, but it was his ideas that attracted her and not the family connection.) In the 1950s Robinson was active in local Democratic party activities, and did less mathematics. She stuffed envelopes, rang doorbells, asked for votes, and so on. She was Alan Cranston's campaign manager in Contra Costa County when he ran for his first political office, state controller.