Ike Closed The Mexican Border

How he did it, courtesy of Christian Science Monitor. The most important element of his plan was breaking the cycle of political corruption by which American politicians (notably LBJ, among others) coddled businesses who profited from the exploitation of cheap illegal workers:

In 1954, Ike appointed retired Gen. Joseph "Jumpin' Joe" Swing, a former West Point classmate and veteran of the 101st Airborne, as the new INS commissioner.
Influential politicians, including Sen. Lyndon B. Johnson (D) of Texas and Sen. Pat McCarran (D) of Nevada, favored open borders, and were dead set against strong border enforcement, Brownell said. But General Swing's close connections to the president shielded him - and the Border Patrol - from meddling by powerful political and corporate interests.
One of Swing's first decisive acts was to transfer certain entrenched immigration officials out of the border area to other regions of the country where their political connections with people such as Senator Johnson would have no effect.