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.
Former Hull City striker Altidore said he did not think about walking off the pitch: "It's a bit disappointing that these things still happen in this time we're in but what are you going to do?," he told the Dutch League's website.
"You just hope that these people can improve themselves. You pray for them.
"I feel like I have an obligation as a football player, to my club, to my family, to not react to things like this and to show that the club stands better than that, that I was raised better than to respond to such ridiculous behaviour.
"It's a bit disappointing. You would hope that we as humanity we can grow from these kind of times but at the end of the day, it's still alive, racism. All we can do is try to educate ourselves, and try to raise young kids to be better than that."
Those responsible had a week to turn themselves in to the club, Den Bosch said in a statement on Wednesday.
"If they don't then we will use the pictures from our security system to find out," Den Bosch director Peter Bijvelds said.
"The perpetrators will receive a ban, while we will also investigate what the possibilities are for legal prosecution."
Alkmaar, who are ninth in the league, were 4-0 up at halftime 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.
Elm praised his team-mate for how he coped with the abuse: "The fans that misbehaved towards Altidore should be ashamed," he said.
"I, like everybody else, am really disappointed in these supporters. It is a bad time for professional football. It is not something I have experienced in my career before.
"Jozy handled it really well. He has not been tempted to respond. He was emotional after his penalty."
On Wednesday, holders PSV Eindhoven entertain Feyenoord and Zwolle host Heracles Almelo while Ajax Amsterdam travel to play Vitesse Arnhem on Thursday.