I would have to say, though I would love to see this happen for various reasons, it isn't constitutional. Bus drivers know when they take a job that they may be randomly drug tested and it is for a reason....public safety. As a nurse, I know that I may be subjected to random drug testing and for the very same reason, my patient's safety.
Welfare recipients do not have other people's lives in their hands, so drug testing has no other reason than to insure their benefits are not being used for drugs.
If that is the reason, then SSI recipients should be subjected to the same treatment. SSI is not a right either....many SSI recipients have not held a job in their life~children, elderly, nonworking spouses, etc. Our taxes pay for both programs, so arguing that our taxes should not pay for other's drug habits is just a knee-jerk reaction to the outrage of the idea of it. But...think of all the prescription drug abuse taking place right now by the elderly and not so elderly people on SSI that we are indeed funding.
Teachers are paid by our tax dollars and all government workers, for that matter....are they all required, before they can pick up their checks, to do mandatory drug testing? Nope. At best they may be subjected to random drug testing but I've yet to see this happen in any school I've known.
If they are going to do this thing, then it should certainly be random and the recipients should know this is a condition of welfare benefits. But, it should also be instituted for SSI recipients, Pell Grant or government loan recipients, etc, as well. If all government assistance programs are going to require it, it should be ALL and not just target one program.