This is a good story, but you can read both the Treaty of Ghent (which ended the War of 1812 while keeping the US-Canada border the same) and Rush-Bagot (which restricted naval fleets on the Great Lakes and Lake Champlain). Neither document states anything about the border needing to remain open or without barriers.

The podcast's transcript suggests that their source for this is https://www.cbc.ca/news/canada/british-columbia/peace-arch-u.... The person confidently claiming that any closure of the park would result in catastrophe is an immigration lawyer, not a historian:

"Saunders said the treaty stipulates there could not be any boundaries or physical barriers erected on the northern border of the U.S. — and if either side violated that treaty — the boundaries revert back to pre-treaty."

Since the Treaty of Ghent restored the pre-War of 1812 borders of both the United States and Canada, this doesn't make any sense. Canadian historian C. P. Stacey states that the period after the War of 1812 actually saw more border fortification than the years before (https://www.jstor.org/stable/1840618).

I mentioned that the idea that a 19th century treaty keeps the US-Canada border demilitarized was a good story, but I think the truth is an even better one: that the border is demilitarized because both countries know that they can trust their neighbor. Let's hope it stays that way.