Well it certainly wasn't James Cook. He was far too late on the scene. Geezers such as Vasco da Gama, Ferdinand Magellan and Cortez were around and sailing south long afore Mr Cook. As to whether it was one of them, however, I don't know.
jno they are allready below the equator so it would be the opposite for them(re the question).but ocean travel tricky one vikings traveled a fair amount and they got a fair distance,but who would know who was first cause they would'nt really know? ,good question.
Vasco da Gama was the first European to round Cape Horn and reach India by sea, that was 1497, 250 years before Cook.
There had been a lot of exploration coming out of Portugal in the years running up to this I don't know if anyone ever recorded who crossed the Equator first because it didn't count for a lot compared to reaching India.
But by the time he did there was already a lot of Arab traders up and down East Africa so they'd probably done it a long time before.
some of the indonesian islands are very close or even are directly on the equator so in theory someone could travel a very short distance and cross the equator. Obviously a name would be impossible to give.
Way before Captain Cook. It could have been the PHOENICIANS, because it is known that they CVIRCUMNAVIGATED Africa, and made their way to GOA in India, for trade.
And one of the Egyptian PHAROAHS actually dug a canal from the Nile River east to the Red Sea, again for trade purposes. Its the Phoenicians for south of the Equator.
And HOW did people get to south America. They were there long before Columbus.