Yes, I am willing to pay $30 for a drug test each time they pick up a check. The cost of the drug test would save thousands in benefits being given to the wrong people. You fail a test? You are ineligible for 3-6 months of benefits. That would easily cover the $30 cost.
The welfare offices were a sign of inefficiency, no surprise there. You could incorporate modern technology into the offices to have less staff. Really, if you make it even slightly inconvenient, less people will make the effort to get the money unless they really need it. Even with the cost of techonology and limited staff, I beleive there would be savings.
There are many jobs brainless people can do. Clean up the roads, summer maintenance, tear down abandoned houses, clean up lots, work in food/homeless shelters. No, I would not trust them with tax returns or drivers licenses.
The bottom line is it is too easy to abuse the system. The more proper channels people have to go through to obtain the money, the more you can cut out the abuse.