In Denmark we tend to have the view that $0 is actually better than poor pay. In a modern country, it only makes sense to put human labor towards fairly high-value purposes, while low-paid jobs are by definition not considered very valuable by the market. Hence we have (roughly, with some asterisks) a $20/hr minimum wage. People who might've worked $5/hour jobs should put their effort towards improving the value of their labor, not towards working low-value jobs. First of all, because low-value jobs are not very useful to society (as a result of not producing much value), and second of all, because if the job isn't paying enough for the person to live on, the government will have to partially support the person anyway, and if the government is supporting them, we'd like them to spend their days improving the value of their labor to change that situation, not spending their time working for a private employer. At least, unless there is good evidence that it's an apprentice-type position that is actually training them, rather than a dead-end low-wage job.

Of course, for that to happen, a system does exist to help people whose labor isn't valuable enough to acquire education or skills necessary to produce more valuable labor. The social system basically takes care of that, paying for education/retraining/apprenticeships, and if necessary covering basic living expenses (rent/food/childcare/etc.) while it's in progress. That can sometimes be done via apprenticeships/internships in the private sector, but with more oversight that they are legitimately training.

So what happens to the $5/hr jobs? If they were really training-type jobs, not much: previously the employer was paying a small wage and the government was basically subsidizing the person's living (because the wage was insufficient to live on), and now that arrangement has just been formalized by making the person be part of a subsidized apprenticeship/training program in which the company pays below-minimum wage and the government contributes the rest.

If it's just a regular job, then if demand is relatively inelastic, and the job is hard to automate, you just pay them more, and the world doesn't really collapse. In effect some money gets redistributed towards lower-wage workers from elsewhere in the economy. If the jobs are easy to automate and worth automating, on the other hand, you just automate them. This is generally good for technological progress, because it pushes the country further ahead on the automation curve. Something that might make economic sense to automate in 2025 elsewhere could be worth automating by 2020 here, because the technology companies don't have to compete with super-cheap labor. It's hard to advance robotics when a human is willing to steal the robot's job by working for a pittance! And contra the Luddites, generally this process improves the quality of jobs available: automating jobs out of existence produces better tech jobs to replace them. Plus, it's going to happen anyway, so might as well speed up the process up by a few years and get out in front of it. One way to do that is to subsidize R&D or pay for trial deployments of new technology, but another way is to just put a price floor on human labor, to discourage the use of legacy manual labor for tasks that should be automatable.

And if the job isn't worth doing at all with either $20/hr labor or however much it'd cost to have machines do it, then it apparently wasn't very valuable! So just do something else instead and no great loss.