James W. Alsdorf, a Chicago business executive and a patron of the arts, died of a heart attack Saturday in Chicago in April 1990. He was 76 years old and lived in Winnetka, Ill. With his wife, the former Marilynn Bruder, Mr. Alsdorf later devoted himself to operating Alsdorf International Ltd., an export and investment business started in 1911 by his father, Anthony James Alsdorf. Mr. Alsdorf was the chairman of the Art Institute of Chicago from 1975 to 1978. He was a founding member of the board of governors of the Menninger Foundation. In addition to his wife, he is survived by two sons: Jeffrey of Seattle and James Jr. of Charlotte, N.C.; a daughter, Lynne of Chicago, and six grandchildren.