Koehler was born in Hamburg; his family spelled their name Köhler until they moved to Milwaukee, Wisconsin in Robert's childhood. Koehler studied art from Henry Vianden and aprenticed himself to a lithography firm. After some time working as a lithographer in New York City, Koehler went to Munich to study fine art at the Royal Accademy in 1873. Koehler's work while in Munich won him silver and bronze awards from the Accademy, and Bavaria's Cross of the Order of St. Michael. Koehler then set himself up as head of a private art school; pupils included Alfons Mucha.
In 1892 Robert Koehler returned to New York City to work as a portrait artist. The following year he moved to Minneapolis, Minnesota, accepting an offer to be the director of the Minneapolis School of Fine Arts. Koehler was also involved with the establishment of Minneapolis' Museum of Fine Art.
Koehler continued working in Minneapolis, painting portraits and landscapes, teaching painting, and arranging exhibitions, until his sudden death from a heart attack at age 66.