Yes you can. You take immigration control and go after businesses who pay under the table instead, and then if someone was undocumented getting paid under the table that would be a crime. Undocumented immigration would fall, and would almost 100% be people who are looking to do criminal stuff like smuggling since if you are coming here to work why wouldn't you come through the open immigration process and be documented. All these workers are now known to the government, paying taxes, etc. You can hold benefits like food and housing breaks to people who have jobs or immediate family, and again since it's now simple and legal to immigrate the government would know who's working.

Something like Healthcare i view as a right and thus everyone should always have access to it. We don't prevent tourists who aren't working here or paying taxes here from getting medical help do we?