That's the reason libertarians want open borders, and that is what would happen if open borders was the beginning and end of how the US dealt with immigration, along with worsening the condition of the places people were coming from.
The proper approach to open borders is to pay reparations to the countries that the US has devastated and ensure workers won't be exploited, whatever their immigration status. Criminalizing human movement to prevent exploitation is not the answer, and it isn't the reason the US has immigration laws now.