The American striker was subjected to monkey chants during the Cup match, which AZ went on to win 5-0 after AZ were reduced to nine men.
Den Bosch director Peter Bijvelds, who had addressed the crowd at half-time in a vain attempt to stop the abuse, said: "FC Den Bosch will, aided by all those people who do good for their club, do everything in their power in the coming period to bring the perpetrators to justice.
"They do not belong in De Vliert (the Den Bosch stadium) and will have the most severe sanctions imposed on them."
Altidore said after the match that he felt it was right not to halt the game and not to respond to the insults.
He said: "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."