Health officials deemed the current risk to public health as low "due to standard infection prevention controls".
All nine mariners provided negative Covid-19 test results before travelling to New Zealand.
The group arrived at Auckland Airport's international terminal from a Southeast Asian country where they were tested again.
Just how long they were inside the airport terminal is being investigated by health authorities.
Other passengers on their plane are considered contacts but as it was a "red flight" all passengers were now in a managed isolation facility where they would undergo standard testing on day 0, 5 and 12, the ministry said.
On the way to New Plymouth, the group stopped at a scheduled toilet stop at a managed isolation facility in Hamilton.
There were no associated locations of interest at this stage.
The vessel had now left New Plymouth and was in international waters off the coast of Taranaki.
All nine mariners were still onboard and isolating.
Health authorities were reviewing the mariners' flight and airport movements as part of the contact-tracing process.
"Due to the standard infection prevention controls and other precautions taken there is minimal additional risk to any of the contacts during the transfer process."
