This question feels as though it's coming from a very modern perspective that really isn't applicable here. You're operating under the assumption that during this period America had fixed, marked land borders with other nation-states and actively patrolled these borders.

To start with, in this period there really wasn't a lot out there. Although all the territory around the nascent U.S. was claimed by some European nation, most of it was extremely sparsely inhabited, whether by whites or native tribes. Going west, you could start in a bustling city and move outward through towns and villages, which would generally grow smaller and farther apart as you went, later giving way to single farms and homesteads, and eventually you would find yourself in the middle of the wilderness, without having seen another human being for days, clearly outside the effective jurisdiction of any U.S. state, but without any idea when you left. So the border as we now know it, with walls and fences or even a marker of any sort did not exist.

Next, a nation is only going to pay someone to patrol the border if they have some incentive to do so. Thus, you see customs officials in all ports above a certain size, whose job it was to visit recently arrived ships and ensure that they paid the import duties on their cargo (tariffs being, in this time, the source of nearly all federal government revenue). But the land border? We must remember that the federal budget in this period, even after adjusting for inflation, was a few tens of millions of dollars- hardly enough to patrol thousands of miles of border, even ignoring the other expenses of the government.

So what you have is a situation with an enormous border that was fixed by treaties but extremely vague in practice which would far outstrip the capacities of the government of the day to patrol, had they the slightest desire or reason to do so.