Domestic US law recognizes that right of asylum. Don't need to bring up any international law.
We have weaponized our incompetence against these people. We're so bad at managing the border that we are unable to process them in a fair, safe, and orderly fashion. We're unable to even follow our own laws and offer the due process our Constitution requires.
The expectation should be that the asylum-seekers do not know the process. Do not know the proper rules. Are desperate and maybe even afraid and just doing their best under life and death (i.e. coercive) circumstances. As the ones who defined that process, it is our job to catch and guide them through it. It is our job to give them their due process. There should be no presumption of their criminal intent. That should have to be proven in a court of law.
It's definitely complicated. Feeding into the right-wing rhetoric about the "illegals" is a way to simplify and thought-kill that complexity.