Further west, the bridges over the St. Lawrence connecting New York and Ontario get lots of daily cross-border workers, usually Canadians working in New York, as well as the normal flow of trade and tourism going both ways. Shutting it down would have a huge impact on the economies of both countries, but particularly Canada.
There certainly are places where you can just walk across a border in northern Vermont, and it's illegal to do so - so not many folks will cross the border.
Remember, we're not at walking-dead levels of societal collapse, we can still generally assume people are following orders given by governments, so just saying the border is closed and letting folks through on a case-by-case basis should be fine.
That is, if Washington State schools are still open.
Still, there are a number of U.S. towns that can only be accessed through Canada.
Also, my relatives in Minnesota say that most of the retail sales in their town is Canadians who don't have anywhere else to shop.
In case anyone is curious, they're referring to Point Roberts, a town whose primary economic drivers are, as far as I've been able to tell, Canadians coming down to pick up packages to avoid cross-border shipping, and providing gas and restaurant meals to those same Canadians.
All public schools in Washington State are closed, FWIW.
There are no border controls when entering Hyder from Canada, as the only way to access the rest of Alaska from Hyder is via seaplane, and those flights are treated as international in origin when they land. There is a inspection station when crossing into Canada, but it is closed overnight -- anyone wishing to enter Canada when its closed must report via videophone.
Edit: I do imagine that many smaller communities close to the border in MB need to cross into the US for necessities, and was being a bit facetious
Edit: I do imagine that smaller communities in MB do actually need to cross the border for a lot of things, but I think Grand Forks has recently been advertising in Winnipeg.