"In a fist fight, we Proud Boys of NZ will f*** you c*** up and i'll smash Valerie Morse myself," the user's post reads.
"That b*** needs to suck out of a straw for a month while in ICU."
The granting of visas comes after Mayor Phil Goff recently banned Southern and Molyneux from using Auckland Council buildings to hold speaking events.
Immigration Minister Iain Lees-Galloway says the decision does not condone the pair's "repugnant views".
"INZ's decision in no way condones the views expressed by the pair, which are repugnant to this Government and run counter to the kind and tolerant values of the vast majority of New Zealanders.
"I understand that many people would prefer it if Lauren Southern and Stefan Molyneux never set foot in New Zealand.
"However, the Immigration Act and immigration instructions have clear criteria for the granting of a visa, including certain character requirements, all of which I have been advised the pair meet."
Neither had been convicted of a crime, nor banned from the United Kingdom or Australia as had been reported, he said.
But Morse said that Molyneux held a classic "neo-Nazi", "white-supremacist" view about certain races of people being genetically superior, while Southern was deeply Islamophobic and intolerant of religious diversity.
"They are people whose ideology is founded in hatred and founded in the idea that there is a sort of war against all going on and that white people have to fight to stay on top of everybody else," she told Newstalk ZB.
"Very serious threats of violence, threats against me to bash my head in, to run me over, this is the kind of violence associated with the speakers."
Morse plans to hand information about the threats against her to police and make a complaint with independent online safety group Netsafe about the social media abuse she has been subjected to.
Federation of Islamic Associations of New Zealand president Hazim Arafeh said the Muslim community was "gutted and sad" the controversial pair had been granted visas.
"We are in consultation now about what is going to be our next move," he told Newstalk ZB.
A group of politicians, lawyers, and commentators, calling itself the Free Speech Coalition, has hired a lawyer to sue Auckland Council for banning the far-right pair from using city-owned venues for their events.
It argues the council and/or Mayor Phil Goff breached freedom of expression under the Bill of Rights Act, and freedom from political discrimination under the Human Rights Act, by refusing to allow the speakers to use the Bruce Mason Centre.
Southern has been hit with a A$68,000 ($74,000) bill by Victorian police for protecting her Melbourne event tonight.
She arrived in Australia last week for a speaking tour in the main centres there.
She told Sky News Australia's Andrew Bolt last night that the police were participants in protests against her because the bill for protection would encourage protests at other events and shut them down.
