He is best known for coaching his children, Croatian skiers Janica and Ivica Kostelić, who won the FIS Alpine World Ski Championships, overall FIS Alpine World Cup and Olympic titles between 2001 and 2014.
Born in Zagreb, Ante Kostelić attended Grammar school in his home town and later graduated from the Faculty of Kinesiology.
In April 2017 it was announced that Kostelić would become coach of the Croatian alpine skiers Elias and Samuel Kolega.
In 2010 he received the Order of Prince Branimir, which is given for excellence in promoting Croatia's international relations.
In addition, the Croatian Olympic Committee elected him several times as Croatia's coach of the year (2002, 2006, 2010, 2014).