It started in with the Immigration Act of 1965. Teddy Kennedy was a big part of getting it passed. It was also part of the Dems idea of making the US look like the world, an outgrowth of the civil rights movement.
The 1965 Immigration Act: Anatomy of a Disaster
1. Teddy was also accustomed to having compromising conversations "off the record" with Soviet diplomats like Anatoly Dobrynin (Soviet ambassador to the U.S.) and Yuri Andropov, when he was General Secretary of the CPSU.
2. Unrestricted Mexican immigration was something LBJ had been fomenting for a dozen years by the time that bill passed. He'd tried to flout President Eisenhower and hold the floodgates open back in 1953-4, but Eisenhower euchred "Landslide Linton" by dealing with the problem via Executive Orders.