If it were me, I would with Mexico’s approval, set up an intake center a few miles from the US-Mexico boarder and have people asking for asylum go there. Those who are relatively safe could stay around there while their claim is processed while those who are at risk will have a section of the facility setup to house them.
It would have an ample number of judges to see the cases in a timely fashion and the food and facilities would be up to at least hospital standards.
If your case is denied, you are sent back to your country of origin or allowed to remain in that area, if you are granted asylum then you are bussed into the US on official transport.
And if your case drags on for more than 4 months, you are automatically transported to a US based facility to continue your case while you wait in America and given a card that will allow you to safely seek employment and also guarantees you 12 months within the US so the employers don’t avoid you because you might not be there for too long. But you will not have any access to welfare as you will still be living and being fed at the facility and not require any of it. As for healthcare, while we are a shithole of a country in that regard but they will be treated same as an American who wonders into an emergency room.