Libertarians would grant amnesty to undocumented immigrants. They advocate open borders and free immigration.
Libertarians advocate free immigration in part because immigration restrictions are highly inefficient. When economists try to measure the deadweight loss immigration restrictions cause, they typically estimate that eliminating all immigration restrictions would double world GDP. That is, we could add another $65 to 70 trillion to the world economy in a few years if only we liberalized immigration laws.
Libertarians do not just say that free immigration would produce good consequences. They also believe that immigration restrictions are pernicious. Libertarians believe immigration restrictions impose poverty, suffering, pain, and death on some of the most vulnerable people in the world.
Philosopher Michael Huemer explains this with a thought experiment: Imagine starving Marvin heads to the market looking for food. Marvin has little to trade. However, suppose there are people at the market willing to trade food for whatever Marvin has. Imagine that unless someone stops him, Marvin will successfully get to the market, make the trade, and eat. However, now imagine that you forcibly prevent Marvin from getting to the market. You post guards to keep him out. The guards continually capture Marvin and turn him away. Marvin can’t barter for food. He starves and dies.