As an undergraduate, Robbins attended Harvard University, where Marston Morse influenced him to become interested in mathematics.

Robbins received a doctorate from Harvard in 1938 under the supervision of Hassler Whitney and was an instructor at New York University from 1939 to 1941.

After World War II, Robbins taught at the University of North Carolina at Chapel Hill from 1946 to 1952, where he was one of the original members of the department of mathematical statistics, then spent a year at the Institute for Advanced Study.

In 1955, Robbins introduced empirical Bayes methods at the Third Berkeley Symposium on Mathematical Statistics and Probability.

Robbins was also one of the inventors of the first stochastic approximation algorithm, the Robbins–Monro method, and worked on the theory of power-one tests and optimal stopping.