You hit on the crux there, whether you meant to or not. All we see and hear are complaints, but not one person offering a solution. Demanding the borders be secured is easy. We all want that. Well, most of us do. How's about some suggestions on how to go about that within logistic sensibility?
"Last but not least,we don't have any idea how many illegals are here. Maybe we should tag the one's we catch like they do some fish."
President GHW Bush was likened to the Antichrist for proposing the exact opposite of that : tattooing legal residents with their SS numbers with ink only readable under certain light. You might not want to push that idea too hard.
Continued and steady enforcement of the existing laws would be a start. That alone would discourage most. It's common knowledge in South America that illegals are ignored - so on they come.