We should impose a must-be-marked-at-all-times lethal health hazard on our own territory for the hell of it? We're not at war here.
I agree doing this over a 2000 mile border is a bad idea. For that, we should try to develop 0-50mi deep monitoring zone (depending on the area), with seismic sensors, UAV overflights, etc., and then whenever anyone is detected, send CBP/ICE agents to intercept. If they're innocent hikers, no harm done; if they're drug/people/etc. smugglers, detain and process them.
That's largely what we do now. The main issue is insufficient CBP agents, and no real consequences for illegally crossing the border ("catch and release"). I think punishment should be much harsher for anyone involved commercially in running the border, and somewhat harsher for people who do it repeatedly.
Ultimately the best solution is to make Mexico a less shitty place to be.