He started at Ohau Point, west of Wellington, about 8.30am and at 1pm was about a third of the way through the marathon swim.
The weather is sunny, with light winds and 1.5m waves.
He hopes to complete the swim to the South Island in about 8 hours.
The 56-year-old said, weather permitting, he would swim from Wellington to the Marlborough Sounds, and hoped to finish the marathon 22km-26km journey in 8-9 hours.
The swim aimed to highlight the critical importance of museums, history and protecting New Zealand's past and the financial pressure that many museums are under in the current economic climate.
He said Otago had an "exceptionally rich history", particularly during the gold-rush era, and it had allowed the museum to acquire a vast and internationally significant collection of more than 1.5 million items.
However, the museum was a charitable trust and only received modest support from ratepayers, and little to no direct central government funding, which left major gaps when it came to essential infrastructure, capital investment and collection care, he said.
One of the most urgent needs at present is the installation of further fire sprinklers to protect the museum's "irreplaceable collections".
"Swimming the Cook Strait felt like something I could do to highlight the circumstances in which the Museum Trust finds itself.
"Water is both the challenge and the solution."
• A Givealittle page to raise funds for the museum has been set up.
