There has to be some way to have an effective closing of the border. But it first requires honest leaders in the majority.
To effectively close the border, you have to remove the incentive to cross the border illegally. One strong disincentive would be the confiscation by law of all property obtained in the United while unlawfully present and employed in the United States, punitive damages for the illegal immigration, and permanent deportation. Prohibition of future U.S. citizenship would also be necessary as a disincentive.