Random drug testing on welfare recipients will end up being more costly than it's worth. Not only would the government need to pay for the testing supplies and lab fees, but it will be costly if they are taken off of welfare as a result of positive test results. Drug addicts don't stop using because they're poor, they will find a way, whether that be theft, prostitution, or other illegal activities. Thus more costs towards law enforcement, incarceration, etc.
Drug testing welfare recipients is expensive. That would be fine if it actually saved us money in the long run. The fact is, when they tried it in Florida, it did not. The number of people caught was so few that they spent more on testing than they saved not paying benefits to drug users.