I suspect, if asked, most would want to close the border down completeley between the legal Ports of Entry, which wouldn't do a darned thing to hinder cross-border commerce.
Speaking for myself, when I say 'close the border', that's what I mean, and I certainly have no problem with people crossing from Agua Prieta into Douglas to shop, or vice versa.
An action along the entire southern border mirroring the current MMP action may have the affect you suggest. But it would only last as long as it took for enterprising businessmen to convince their clientele that it really is okay to come back and shop.
And in my book, that's a win-win for both sides.
There are many business and civic leaders who would dis-agree with you.