Two police cars and several officers are at the scene, near the Interislander ferry terminal.
The three remaining activists are standing beneath a banner saying "Climb It Change."
A Greenpeace spokesman said the activists had several hours worth of food and drink and were experienced, well-equipped climbers.
The activists intend to stay as long as possible.
Russel Norman, Greenpeace New Zealand executive director, said the activists were undertaking a non-violent direct action to raise awareness about climate change and the ongoing extraction and burning of fossil fuels.
"The alternative is renewable energy."
Dr Norman said all activists were briefed, and knew arrests were likely.
"Obviously we'll support them."
Greenpeace said Tangaroa was refitted at taxpayer expense to exploit hydrocarbons in an operation likely to undermine New Zealand's obligations to fighting the greenhouse effect and climate change-related issues.
Greenpeace said the taxpayer-funded boat had been surveying the East Cost of the North Island on behalf of petroleum giants Statoil and Chevron.
The protesters are now preparing to uncurl a sail-shaped banner from the mast with the words, Climb it Change, while the others have attached others all over the boat.
The ship had only been in Wellington Harbour on a pit stop before it continued with its oil exploration venture.
The protest comes days before the Climate Conference in Paris, where Prime Minister John Key will be one of the many world leaders attending the forum discussing how to tackle climate change.
Greenpeace climate campaigner, Steve Abel, said Mr Key's government was "dishonest".
"Right now, as John Key gets ready to head out to Paris...this science ship that should be doing vital environment work is trying to head out to survey our waters for the climate-wrecking oil industry."
He said oil was destructive to the planet.
"This will cause global suffering, including for thousands of families in New Zealand and the Pacific who could be forced from their homes by extreme weather and rising seas."
He said the protest slogan was a nod to civil disobedience and protestors efforts against the fossil fuel industry.
Today's action is just the start protests this week, with the People's Climate March gets underway this weekend.
People from all over the world will be gathering in numerous cities to call for action against climate change.
"We need to take the power back and protect our children's future," Mr Abel said.
