While I have no idea how much unemployment fraud actually occurs, it is rather easy in many states to get away with it. They just ask you where you applied and that's it. You are given a warning that lying on the form is punishable to x extent by the law. I don't know what percentage of these are actually checked.
Drivers licenses have numbers and barcodes. Why not have a computerized system that's harder to defraud? Instead of asking where they applied, have their drivers license(or state ID) linked into the system such that it can be scanned upon application? Many applications already ask for the number for background check purposes. This would be easy to implement as most places of employment have scanners, a computer, and/or online applications.
Businesses and employees are assigned numbers for the system.
Online applications could ask for the number(many already do) and put the number into the system. The number assigned to the business is then added to the person's file in the new unemployment system.
Upon finishing an application in-store or delivering a finished application to the store, a drivers license or state ID could be scanned or the number entered manually. The number assigned to the employer and the employee are then added to the person's file in the unemployment system. For stores that somehow have no internet access, there could be a paper form to be filled out with the application that is then mailed to the unemployment agency with the name and signature of the employee and the applicant.