For the most part we do. The current angst is over asylum seekers, not workers.

Anyone who wants a visa to work here can typically get one. Whether that is a professional (H1-B), farm worker (H-2A), seasonal resort worker (H2-B), Exchange worker like an Au Pair (J), professional from Canada/Mexico (TN), Athlete (O), investor (E), etc. Although there is a process to go through, just about anyone who wants to work in the U.S. can do so.

Issues arise when a Visa is over stayed.