How is this issue solved by increasing liberty instead of decreasing it?
Well, the first thought that comes to mind would be considered somewhere between "racist" and an act of war... and involves bubbas, beer, and hunting rifles...
nevermind that.
Seriously,
If, by enforcing existing laws which prohibit incursions into the US, and securing the border through physical means to better control the movement of arms from the US into Mexico, the problem is Mexico's from there. That might have the additional benefit of hampering the flow of illegals and drugs from Mexico.
That would remove the onus from the US, and preserve, rather than increase Liberty. In this case, a stalemate is almost a 'win'.
Unless the Mexican Government desired to mount a joint military operation against enemies of the Estados Unidos Mexicanos
and the United States of America along our mutual border (the Cartels), I don't see how it would be otherwise.
I am not sure how the US military could be legally used in an operation which would flirt with violating Posse Comitatus (18 U.S. Code § 1385),
https://www.law.cornell.edu/uscode/text/18/1385 without a declaration of war against the drug cartels, which, imho, is where the drug war should have been fought all along.
They are worse than the Barbary Pirates, yet we fight on our streets instead of the source.
They are notorious for barbarity, and will not go quietly, up to and including acts of terrorism.
The other side of that is the potential for confiscation of assets which could be used to build that barrier or offset the cost of operations. That, too, gets into muddy water, legally with the potential to develop agency robber barons who would look for assets to seize, more than illegal activity.