The border patrol approach will never work any more than raids on puppy mills change the economics of the puppy trade. Even based on workplace enforcement, however, there are no easy answers. For years I owned and ran a business where a significant portion of our staff were foreign nationals. We always verified visa status, but that does not mean that we never employed illegals. The problem is that status can change and the employer has no reasonable way of knowing that. In addition, we have no way of identifying papers that might be fraudulent. We were no more able or interested in playing the job of immigration policeman than the hospital, the local police officer, the man at the newsstand, or you. And in case you don't know, it is illegal to discriminate in employment based on national origin as long as the person is legally eligible for employment in the United States.
However, as long as illegals are here, they will still need medical care. The care they are receiving now is generally only for medical emergencies. It may be for treatment of a contagious illness. It may be for treatment of a life threatening emergency. I assume you are not suggesting that hospital emergency rooms should withhold treatment until you can produce both an insurance card and a passport and visa. As of now, we are still not branded with our permanent citizenship number at birth.
Most local police departments have enough to do fighting crime in their own communities and do not become involved in investigating immigration status. Hospitals are even less interested in investigating alleged immigration violations. State Medicaid programs often provide coverage for illegals because the alternative is that the bills are unpaid. In Pennsylvania, the state provides annual payments to hospitals for the unreimbursed care they are legally required to provide. The balance gets built into provider rates paid by patients and their insurance companies.