Visa overstays are a problem because the are an exploited loop-hole in our immigration system.
America should be able to tightly control who comes and goes from our country.
I would propose limits on the visas we offer to a foreign country. Such that the number of over stays is decremented from the visa allotments allowed for the country (or we stop issuing visas to that country all together) and that we impose hefty fines on the country for everyday the visa overstay remains in our country past the visa expiration. In addition, if any resources are used to locate and remove a visa overstay we should be able to recuperate those costs from their home country as well.