If they had a choice, airports and airlines wouldn't pay for passport control services, because they'd rather not have passport control, it adds confusion and delay to the travel process.
Since governments generally insist on having these services, they should be the ones to pay.
I don't really see how is this a question at all. The state mandates passport control, and any airport wishing to operate legally must contract the border patrol for this, and pay for them. And it should be per checked passport, and they should be able to contract staff based on the predicted demand curve.
If it's a new private highway, then the operators should, sure. Other than that it doesn't really matter, as Congress probably budgeted for the border crossing when it decided to build the highway and connect it to the neighboring country's highway/road.
If they had a choice, airports and airlines wouldn't pay for passport control services, because they'd rather not have passport control, it adds confusion and delay to the travel process.
Since governments generally insist on having these services, they should be the ones to pay.