I will quote a post I wrote where I touched on Kurlansky's work:

Mark Kurlansky in his pop-history (non-academic with no citations or references) works Cod and Basque History of the World for some reason strongly proposes the theories of Basque knowledge of North America, himself confesses the following in Basque History of the World:

The two leading arguments for placing the Basques in pre- Columbian America are both based on deductive reasoning.

and

But no physical evidence has been found of the Basques in North America before Cabot. Historians and archeologists who have searched for it and failed insist that the rumors are false. But the search for pre-Columbian Basques in America has yielded ample evidence of a surprisingly large-scale Basque presence in Newfoundland and Labrador soon after Cabot. The remains of extensive Basque whaling stations dating to 1530 have been found.

As we see, he admits that there is no physical evidence and proceedes to demonstrate his case on deductive reasoning, and honestly, it rests on some really eager jumps to conclusions and some really bad historical premises. For example he says:

But the Basques chased whales that traveled to subarctic waters and then dropped down along both the European and American coastlines.

But to quote an academic work about the Newfoundland fisheries "The Basque Whaling Establishments in Labrador 1536-1632 —A Summary" by Selma Huxley Barkham:

Contrary to the spurious claims of writers on the history of whaling who have based their findings on secondary evidence, the Basques never, at any point, chased whales further and further out into the Atlantic until they collided with North America. This ridiculous legend must be laid to rest once and for all.

Kurlansky book has some other mistakes and misleading statements in that chapter that would warrant a good badhistory post for itself

So in short, Kurlansky's argument rest on no evidence whatsoever, but mainly "deducing" from various other information, and even that is based on some faulty premises and reasoning