What does unemployment insurance have anything to do with it? That's not welfare.
Who the hell cares if you are a drug addict or alcoholic if you are getting an UI check???
It's like saying they should screen you for drugs and alcohol before you can get an insurance check if your house burned down. You paid into it and if you need to use the insurance then you should damn well get it.
You have to meet certain criteria in order to be eligible for UI, making it somewhat similar to welfare. I don't claim that it's the same thing, but there are similarities between the two.
My point still stands... people lie, and they can certainly lie about their drug and alcohol usage when applying for welfare.