No cash, no open borders - simple. Even a caveman could follow the money ... pay the politicians and guards and presto your theory comes true - however, do not pay the politicians and the cartels will not be into people trafficking and there would be a fence - a big one. Maybe with tower mounted motion sensing .50 cals ...
Well, ummm, no.
The US border has been wide open for decades and will continue to be so without any new political money (not counting cartels and Mexican politicians).
Democrats will continue and always oppose any check on illegal immigration for votes; and Republicans, with few exceptions, are too balless to oppose it. No further money required to sustain.