Referee Reinold Wiedemeijer spoke to club officials when the chanting broke out in the first half and instructed second-tier Den Bosch to put a message over the public address system, warning the crowd he would stop the match if the abuse carried on.
The tie was briefly suspended early in the second half after misbehaving home fans threw snowballs at the linesmen.
The quarter-final then resumed and top-flight Alkmaar went on to complete a resounding victory as the home side finished with nine men following the sendings-off of Jelle de Bock in the 38th minute and Joey Brock in the 51st.
Alkmaar, who are ninth in the league, were 4-0 up at half-time thanks to goals from Viktor Elm, Johann Gudmundsson, Adam Maher and a penalty by Altidore.
Maher added his second goal, and Alkmaar's fifth, on the hour.
On Wednesday, holders PSV Eindhoven entertain Feyenoord and Zwolle host Heracles Almelo while Ajax Amsterdam travel to play Vitesse Arnhem on Thursday.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.