Born in Bussum, North Holland, Hesp started his career in the 1985-1986 season at HFC Haarlem. After two years where he rarely played he moved to another club in the Eredivisie, Fortuna Sittard, where he would remain the following seven years. During his first ten professional campaigns (one with Fortuna was spent in the second division), he did not miss a single game.
After three seasons with Roda JC Hesp signed with Spanish side Football Club Barcelona, as the Catalans were being managed by countryman Louis van Gaal, also newly-signed.
In his last season at the Camp Nou Hesp split first-choice status with youth graduate Francesc Arnau, then moved back to the Netherlands and Fortuna, retiring at almost 37 in 2002. Subsequently, he joined Football Club Groningen as a goalkeeper"s coach.
In the 2013 summer Hesp left Groningen for PSV Eindhoven, in the same capacity. He also worked with the national team as a goalkeeper coach.