Looks like Cadet Jack has the floor.
The answers (though without ever getting an admission from Roddenberry, they may only be opinions) to the original question were Banks and Endeavour.
The ship was (the Airfix) Endeavour, James Cook was the inspiration for James T. Kirk and Sir Joseph Banks the model for science officer Spock.
The "prime directive" mimicked orders given, not only to Cook, but later Pacific explorers De La Perouse and Malaspina, to establish good relations and to use violence only as a last resort, with any of the new civilizations they encountered.
The Endeavour, Resolution and Enterprise were all on multi year missions.
On his second voyage, Cook claimed that he had sailed "..farther than any other man before me". Kirk claimed "to boldly go where no man has gone before"
Cook's logs of all three of his voyages were international best sellers... Kirk opened every episode with "Captain's log: stardate..."
Cook landed with his red coated marines on new unexplored islands, Kirk transported to new unexplored worlds with red uniformed (usually expendable) security details.
Cook's men were greeted by sexually uninhibited Polynesian babes... Kirk had the green babe, among others... remember the androids?
Kirk's ship was overrun with "Tribbles", Cook's ship was overrun with giant cockroaches picked up in New Zealand.
A common theme in Star Trek was the "wow" effect the trekkies had on technncally inferior civilizations...just like cannon, muskets and fireworks impressed stone aged islanders.
Way too many parallels for coincidence, except of course, Kirk never got to be the main course at dinner.
By the way, a Lt. Bligh was Cook's sailling master in Resolution.