"The association was made aware of an allegation involving Mr Johnson and raised it with him. It has since accepted his resignation," said Dykes.
"This has been a very sad and disappointing episode for all concerned."
Johnson, a highly regarded coach who has worked with New Zealand Cricket in the past, issued a statement in which he apologised for his conduct.
"I have made some very bad decisions around my personal finances which have compounded to a point where I have let down Otago Cricket and myself," he said.
"I sincerely regret my actions and must now concentrate on restoring my health so I can do my best to make amends."
Dunedin-born Johnson, a former seam bowler who took 64 wickets in 27 first-class games for the province, and played 20 one-day games, had been on his second coaching stint at Otago.
He also spent 2000-06 as Wellington coach and has filled coaching roles with New Zealand Cricket since the early 2000s.