Doctor Dobson received his MD degree from the University of Chicago School of Medicine. He obtained his dermatology training from Dartmouth-Hitchcock Medical Center and served a research fellowship in the United States Public Health Service. Following teaching appointments at the University of North Carolina and University of Oregon medical schools, he became Professor and Chairman of Dermatology at SUNY-Buffalo School of Medicine, serving for over 7 ½ years until 1980. He was then appointed to the professorship and chairmanship at the Medical College of South Carolina in Charleston, where he is now emeritus professor and for whom a named lectureship has been established.

He has received many accolades for his teaching and research that has often been focused on the eccrine sweat gland. Doctor Dobson was the second Editor-in-Chief of the Journal of the American Academy of Dermatology and has been President of the American Academy of Dermatology and the American Board of Dermatology.