The thought of us doing this to our own territory is ridiculous.
A well-marked, time-limited or centrally controlled minefield is generally much less dangerous, on par with a razor-wire fence. Functionally, it's a sentry system which can be turned off electronically (or by waiting past a certain date), or at worst, which is marked on maps well enough that individual mines can be removed safely by engineers.
Unexploded ordnance (i.e. bombs which you fire and which then fail to explode) are a bigger threat after a modern battle than US-style minefields.
A minefield is probably much less likely to be deployed for this than automated sentry towers, however.
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.