Radiolab (the podcast) did a good series on border policy that was pretty informative.
IIRC, the US-Mexico border was never open, but it was quite porous. People would come to the US, work for a season to make some money, then go home.
Then once border security started to be tightened, leaving was too risky so people would stay leading to a huge increase in the number of undocumented people living in the US.
To expand on this. California has the majority of illegal migrants. A significant portion of them are "seasonal" fruit pickers where there exist some migrant farmer visas for them, but not nearly enough. Now should we have American citizens/PR holders doing these jobs? Maybe, but it's pretty hard to get Americans to do this back breaking work in the hot sun for a few weeks, travelling from farm to farm until all the fruit is picked. American fruit is really only as affordable (and available) as it is thanks to all this migrant seasonal labour.
I think there's probably some leftist argument against this system on account of how we're taking advantage of poor countries and something something workers rights, but I think removing the ability to leave, to go home is the worst of both worlds.
IIRC, the US-Mexico border was never open, but it was quite porous. People would come to the US, work for a season to make some money, then go home.
Then once border security started to be tightened, leaving was too risky so people would stay leading to a huge increase in the number of undocumented people living in the US.