There are a few simple things:
1. First, let it be known it is a crime to be here illegally, one punishable by prison or deportation. Enforce existing law.
2. Cut off all social services and welfare to those here illegally.
3. Enforce large sanctions and penalties against those who hire people without proof of citizenship or legal work Visa. Make certain the punishment for overstaying your Visa to be here is a strong one.
4. Fashion any new laws concerning illegal immigrants with an eye of getting them to self-deport.
5. Stop chain immigration through family ties.
6. Change the law concerning anchor babies.
If these kind of priorities were put in place, the problem would be quite small.
The only medical care should be emergency or first aid sufficient enough to get them back home in their country