To some degree, it is a government problem. But adding more intervention, in the form of tariffs and high taxes, will probably not help. Ideally, the government should not be involved at all, whether it is by issuing special visas or blocking entry to the country to protect "American" jobs.
If not for the health risks and tremendous security risks, I would favor open borders. I'm realistic enough to see the need to regulate entry into the country to make sure someone isn't smuggling in a nuke or a deadly contagious disease. Government has a legitimate role in defending the nation, so borders and border guards make sense.
But I don't think that immigration regulations should extend to distorting the labor market. No Amercian has the "right" to a job. And I'm saying that as someone who lives in Texas and faces the prospect of competition from immigrants from all over the world.