Morality is irrelevant; minimum wage is an efficiency model. More to the point, though, minimum wage espouses a particular goal, and it cannot meet that goal if its purchasing power becomes continuously lesser.
If it were about morality, we'd have an unresolvable conflict: implementing a minimum-wage increase throws some of the poorest of poor out into the unemployment line to starve; while not implementing a minimum-wage increase lets all of those poorest of poor continuously face greater hardship until they begin to starve. QED, who gets randomly executed because fuck it?
Your labor force is made of adults. Children don't produce; they simply consume. Your laborers work for forty years--from age 18 to age 58, roughly, although it's longer now--and if one of them dies, you need eighteen years to begin replacing them. That assumes you can pop out a baby, feed it, clothe it, give it medical care, grow it to an adult, and then dump it right into a job without investing any more in preparation which you could have avoided by not killing your previous laborer. The other side of this is we expect retirees to be essentially cheaper than children, or at least we want the full ROI of their employable lifetime before they become an economic burden.
Minimum wage is a type of welfare. We have a minimum wage and public aid system, which ensures that the working-class at least get a minimum viable income, while the reserve labor force (the unemployed and underemployed--not working full-time) gets aid to keep them alive and healthy. It's spotty, and worked as best anything could before a Universal Social Security became technically and politically viable; now the United States can now end all hunger and homelessness at a $1 trillion reduction in total costs to the taxpayer--without raising taxes on anyone--and so that's technically-better.
These aren't feel-good moral actions. These are efficiency. When you come up short on efficiency in an economy, people die unnecessarily. You have the capacity to care for the sick, to feed the hungry, to supply the means to live, to stabilize lives; and you squander it, you waste it, and so people die of disease, they starve, they become homeless. The more-efficient your economy is, the greater the standard of living; and the more-stable your economy is, the less-likely people are to have good savings, a good income, insurances, everything to keep them ready for any sudden life crisis and still suddenly end up poor, homeless, and dying of diseases we should have eradicated decades ago.
We trade efficiency away for moral reasons. We accept more death, more poverty, and more suffering so that we can pursue things which we enjoy, and so we can go through life without living in constant fear. The ideals for efficiency by central command simply don't work; but efficient control through constant surveillance, state-controlled information to shape political opinions, and other means of crushing out freedoms can bring strictly-better prosperity. That's bland and it takes away peoples's humanity, or something along those lines; it's the kind of world nobody wants to live in unless they're in charge--and often not even then. Those are the things of which we accept the costs, although to be fair they're usually costs paid by someone else--most of us don't end up the one starving in the streets because of a little loss of efficiency, so we'll gladly trade it away.
You can't claim "morality" if you're going to be blind to what pain and suffering you do and do not cause taking your high ground. That gets you such brilliant, morally-sound ideals as cutting off trade with China, condemning hundreds of millions of people to joblessness, homelessness, and starvation, because we think their wages are too low and want to equate Chinese labor to slave labor. Murder on a grand scale far beyond anything Hitler ever did is what a surprising number of people believe would be "morally-correct" and "The Right Thing To Do(TM)", so long as they can stand far enough back from the carnage to claim their hands are clean.