Born in Palmyra, Iowa, Craig was educated at Iowa State University where he obtained his PhD in 1931. After two years at Johns Hopkins University he moved to Rockefeller University, New York, in 1933 where he was appointed professor of chemistry in 1949.

Craig concentrated on devising and improving techniques for separating the constituents of mixtures. His development of a fractional extraction method named countercurrent distribution (CCD) proved to be particularly good for preparing pure forms of several antibiotics and hormones. The method also established that the molecular weight of insulin is half the weight previously suggested. Craig also used CCD to separate the two protein chains of hemoglobin.

During work on ergot alkaloids Craig, with W. A. Jacobs, isolated an unknown amino acid, which they named lysergic acid. Other workers managed to prepare the dimethyl amide of this acid and found the compound, LSD, to have considerable physiological effects.