It's understood the NT police Territory Response Group was sent to patrol the house on Saturday night, but the Chief Minister would not comment on the security details.
"I think people understand for good reasons I can't provide details on what the security precautions are or have been to make people safe," he said.
"We don't want people knowing how we're making sure that myself and my family are safe but obviously precautions had to be taken because of the threats that were made and my address was given out - and not just given out, people were encouraged to go to my house."
Police say at least 1000 people were at Saturday's rally, while organisers believe up to 3000 people attended.
Anger spilled over after the NT Government announced mandatory Covid-19 vaccines for anyone working in contact with a vulnerable person.
The new rules that make it mandatory for many workers who interact with the public to have their first jab by November 12. Second doses will need to be complete by December 24, otherwise they face fines of up to $5000.
The move has attracted negative attention from around the world.
US Republican Senator Ted Cruz weighed in on Twitter, describing the policy as "Covid tyranny" and saying it was "depressing and sad".
Gunner responded to Mr Cruz, defending his approach to Covid, in a post that has gone viral.
"Hey Ted Cruz, g'day from the Northern Territory in Australia. Here are some facts. Nearly 70,000 Texans have tragically died from Covid. There have been zero deaths in the Territory. Did you know that?" he wrote.
He went on to spell out that vaccination was key to the protection of vulnerable Aboriginal people who belonged to "the oldest continuous living culture on the planet".
"We've done whatever it takes to protect the Territory. That's kept us safe AND free. We have been in lockdown for just eight days in 18 months. Our businesses and schools are all open. Did you know that?" he said.
"We don't need your lectures, thanks mate. You know nothing about us. And if you stand against a lifesaving vaccine, then you sure as hell don't stand with Australia. "I love Texas (go Longhorns), but when it comes to Covid, I'm glad we are nothing like you."
Gunner's tough vaccine rules have also been met with local opposition. The Country Liberal Party has described it as "draconian", while Federal CLP Senator Sam McMahon likened it to Nazi Germany.
However, Gunner is not backing down.
"You have a choice, if you want to work in a setting that interacts with vulnerable people, a classroom with kids who can't be vaccinated, you can choose that, I'm asking you to choose to be vaccinated," Gunner told Sky News Australia.
"That child doesn't get to choose to be vaccinated. An Aboriginal person in a remote community doesn't choose to be vulnerable. You get to choose whether you work in that setting or not and we're saying the health advice here is very clear that with the Territory and our vulnerable settings you need to be vaccinated if you want to work there."
