Why do you think the borders are open?
Even assuming you have a good answer for that, what makes you think it matters? Birth rates are falling globally. Open borders only forestalls the inevitable.
Coming to the border, making a (legitimate|illegitimate) asylum claim, being processed, given a Notice to Appear, and being released into the interior of the United States is as easy as a $1-2,000 payment to smugglers.
From there, either working under the table or working off of a stolen identity is the norm. Thankfully, the latter is hard because the United States has a robust identity verification system with the secure, secret identifier known as the Social Security Number and thankfully all states require employment verification to prevent abuse of stolen identities and the abuse of these workers.
People are just walking over the border nowadays, not sure if smugglers are even needed.
I don't think you can say that that's "nearly impossible", and even if it is, the numbers are still very large; by far the most in the world.
That said, this shrinking growth is a worldwide issue. Unless America imports more Africans (again...) and other 3rd world countries to try and balance things out, it's going to be a rough gen Alpha and beyond.