He's not saying you have to be paid UI for any reason. This is another reason we shouldn't be introducing new, ridiculous standards that are not going to be fairly enforced. The requirement of unemployment is that you are available to work and looking for work. That's all that should matter. Whether a person is able to survive until they find a job (and hopefully sooner than later) should be the goal of Unemployment Insurance. Anything that claims to help us find criminals during this kind of process is really only going to help make government more intrusive and expensive.
Firstly, some states do have a UI tax because this is a state administered program that is often bailed out by the Federal government as it is being now. And also, If you are a conservative or libertarian, it should be common sense to you that employers consider these taxes a cost of hiring the employee that they factor into their value as a new hire.
Sure, you can make the argument from the left that they wouldn't necessarily offer them the same amount of money in a totally free market, but the good chance is there since they are willing to spend it when they have to.
And as an aside, when I was an employee of the State of North Carolina (working in a somewhat political capacity) I was never once approached by any person or document about drug testing, random or otherwise.