the Romans developed this very problem when they went from active border maintenance to defense in depth.
it resulted in the eventual de-romanization of the provinces, and the eventual crumbling of the western empire.
we are fools if we characterize this as a minor problem.
The US hasn't had much of a border until recently. The empire had a frontier up until 1890, officially, but population was low enough that the border economy wasn't capable of attracting wetbacks in any quantity and any and all were welcome to settle in the frontier. I wouldn't propose that we form a solid border even now. America is more a state of mind than any buildings and other piles of stone. If the Terrorists and more formal enemies start blowing up the buildings, they won't touch us except to incite the relatives and friends and neighbors of the injured to demand revenge and victory over this scourge. Mexicans? They are a lot like us and will band together with us.
BTTT
Very good observation. To the Romans their DiD made military sense, but in the end they paid the price.