Why do you want to sentence people to working menial jobs which could be automated away?
It's like trying to encourage employment of human alarm clocks or switchboard operators.
Because the mindset that people need to work in order not to be thought of as unproductive leeches is not going away; if anything, it's become worse. The result is that people who lose their jobs and cannot find other employment through no fault of their own are thrown into the underclass and given barely enough to survive. Furthermore, not being employed has severe negative effects on mental health. If there were a serious chance of a change of mindset, I'd rather just dial the average workweek way back and have everyone reap the benefits of the extra leisure time. But this isn't what's happening: instead, people who have jobs work full weeks, and people whose jobs have been eliminated get discarded.
If you read things written by social reformers and futurists of the turn of the 20th century, they agree that the increase in productivity caused by automation should cause the average work week to fall to something like 20 hours or less by the year 2000, thus freeing up people's time for leisure, hobbies, further education, etc. But there's no sign of this, and it doesn't appear that our economic system favors decreasing the hours of people who are employed in order to employ more people. Instead, people who do have jobs work 40+ hours/week, and people who work less or not at all are usually working less because their employer cut their hours and their pay, or laid them off altogether.
So we're in a situation where we basically need people to dig holes and fill them back in again, or else they'll be discarded as unproductive leeches. Well, better would be to employ them at some of the things we actually do need, like infrastructure repair and upgrading, but there will come a point where it's not possible to productively employ all of the workforce (minus a few percent unemployed at "full employment"). This will cause a long-term increase in long-term unemployment. We're already seeing this today, although it's hard to tell where the technological effects begin and the normal economic ones end.
What you are describing has been operative more or less since the beginning of the industrial revolution, and even earlier. Marx recognized it all the way back in the 19th century.
But, I think you're completely off-base if you represent the minimum wage level as having a major effect on the processes that cause this to happen, considering how few workers are actually affected by the minimum wage.
In minimum wage discussions, you have to consider not just the workers who are working at exactly the current minimum wage but all of the workers who are currently making wages between the old minimum wage and the new one that is being proposed. According to metalhead's source, that's something like 40% of the workforce between $7.25/hr and $15/hr.
It's nonsense when it's used to object to raising the minimum wage to $15 over the next four years.
Where is the actual evidence that minimum wage increases really cause these effects?
In theory, sure, you could raise the minimum wage to $100 or more, but is that even remotely on the table? No.
http://www.npr.org/2016/04/01/472716129/one-year-on-seattle-explores-impact-of-15-minimum-wage-law
There haven't really been any good examples where the minimum wage was raised to anything like that level, even over four years, while an adjacent jurisdiction didn't raise it. That's the main reason I'm excited that some cities are running the experiment.
Even for the usual kind of minimum wage hike (from something extremely low to something very low), economists are approximately evenly divided on whether they increase unemployment or not. My best understanding of the evidence is that the normal type of minimum wage increase doesn't have any real effect on employment, but a large one (say, hiking it to the current 40th percentile wage) nearly certainly would. Section II of
this blog post contains a decent summary of the disagreements among economists.
My guess about the Seattle case is that Seattle will be able to get away with raising its minimum wage to $15 without major ill effect, as could some other high-priced cities with strong economies on the West and East coasts. However, it would almost certainly be bad policy in most of the country. Even within states, a gradual hike to $15 in Washington would have vastly different effects in Spokane than it would in Seattle, or in California Fresno vs San Francisco, or in Massachusetts Fall River vs Cambridge, etc.