The people were on a $21 million boat owned by Zuru founder Nick Mowbray.
Mowbray told the Herald he was "obviously really disappointed to learn about these insensitive videos".
"It's the complete opposite of what I stand for and what Zuru stands for. I can't speak on behalf of the people who were involved, but I always encourage anyone who is called out for this type of unacceptable behaviour to take ownership, apologise and learn from their mistake," Mowbray added.
The people tagged in the video appear to have since deleted their social media accounts.
Māori Party co-leader Debbie Ngarewa-Packer said she was "disappointed" with the people in the video, who she at first believed to be foreigners "who didn't know any better".
"I think it's a sad reflection on what should otherwise be a fabulous race," she added.
"It's important to remember that the America's Cup has been supported widely by tangata whenua. The name of the yacht - Te Rehutai - the name it's been blessed... We believe it's our karakia that helped it win.
"It's a disrespect of the hard work that has been going on to make this unique Aotearoa event," Ngarewa-Packer added.
"Sadly I think it shows the ignorance of those who haven't taken on board or respected the way that Māori culture and New Zealand as a whole has wrapped around this event. The tangata whenua there have been hugely generous. Named the yacht, and, more than a stamp of approval, it gave it the stamp of absolute protection."
The Māori Party co-leader said the footage taints what should have been a day of pure celebration.
"It's 2021 and people expect a better degree of behaviour than appropriating a culture in a derogatory way," she said.
"I'm sure there are people in Tāmaki Makaurau who would be willing to teach them better way to celebrate the culture, if that's what they were trying to do."
The acclaimed DJ, real name Boris Daene, posted an apology on Thursday morning, after deleting the video.
"I'm embarrassed and ashamed of myself. I obviously have a lot to learn. I need to educate myself in issues of cultural appropriation and I am committed to doing this. I apologise to everyone I've offended," he wrote on his Instagram Stories.
He posted the same statement to his Twitter account.
Over the course of the day, numerous people have taken to social media to condemn the video, which was shared to Twitter by former gossip columnist Pebbles Hooper.
It was then reshared by multiple people who said the people in the video were mocking Māori culture.
The party took place on Mowbray's boat, a 40m $21 million Princess superyacht, which he purchased late last year and brought to New Zealand from the Mediterranean.
The boat boasts three decks, a spa and entertaining rooms, and five luxury cabins with en suites.
The incident also reminded social media users of a separate incident earlier in the America's Cup, when American Magic fan Ollie Wall, son of legendary real estate man Graham Wall, dressed up as horned Capitol rioter and QAnon supporter Jake Angeli.
Back in January, Wall was photographed yesterday posing as the figure known as "QAnon Shaman" on another boat with fans watching the Prada Cup races on the Waitematā Harbour.
He then apologised for what he described as a "silly outfit choice".
