Standard Infection Prevention and Control (IPC) protocol, including the appropriate use of PPE, was used throughout the crew's testing.
After the tests were taken, the ship left Port Taranaki and is now on its way to Lyttelton where the appropriate level of support can be provided.
"Health authorities are working on arrangements for the crew once they arrive at Lyttelton Port," the Ministry of Health said.
"This includes whether they quarantine aboard the ship or are transferred to a facility on shore. If any confirmed cases are to be transferred, that will also be done using standard IPC protocol."
The Playa Zahara's shipping agent and its crew are cooperating with officials.
Whole genome sequencing is under way.
Port Taranaki said it was satisfied with all safety precautions that were taken when the vessel was docked at the port on Tuesday for crew members to undergo Covid-19 testing.
"No Port Taranaki staff or other port workers went onboard the vessel or had contact with those on the vessel," Port Taranaki chief executive Guy Roper said.
"The vessel was remotely piloted in and out of port, and those tying up the vessel wore full PPE and had no contact with crew.
"The vessel moored at a special area set aside for quarantine vessels on Moturoa Wharf, and the wharf area was washed down when the vessel returned to sea."
Roper said the port had supported the Ministry of Health's procedures "to ensure that the safety of the community and the welfare of the crew has been the top priority".
On Tuesday, Roper said the vessel had been at sea for more than three weeks, so there has been no contact with others or any crew changes.
No Port Taranaki staff members would be in contact with those on board the vessel either, he said then.
"Should any crew members return a positive Covid-19 test result, we expect the vessel will head to a port where there are managed isolation facilities close by, as has occurred with the Viking Bay," he said at the time.
The Viking Bay was the vessel that first reported a Covid case among its crew, with two mariners confirmed as having Covid-19 last week.
A further 13 crew members tested positive this week and have been transferred to an on-shore managed isolation facility in Wellington. Both fishing vessels are Spanish-flagged.
The two vessels are not fishing in New Zealand waters and are not associated with the New Zealand fishing industry.
A spokesperson for Lyttelton Port Company referred the Herald to NZ Customs, while Taranaki District Health Board referred enquiries to the Ministry of Health.