Cucolo was born in Suffern, New York on August 22, 1957. Cucolo's grandfather, Anthony Sr. (1897-1983) was an immigrant from Italy who rose from laborer to president of a half dozen companies involved in the construction business. The senior Anthony Cucolo also served as a captain in the U.S. Army's Engineer branch during World War II. Anthony Cucolo III's uncle Belmonte Cucolo (1920-2005) and father Anthony A. Cucolo Jr. both attended the United States Military Academy, graduating in January 1943 and June 1946, respectively. Belmonte Cucolo served in the Pacific theater during World War II and attained the rank of captain before leaving the Army in 1947. Anthony Cucolo Jr. served in Korea during the Korean War, and attained the rank of captain in the Infantry branch before resigning in 1954.
