There is one problem with this debate that I am seeing: People are just assuming that illegals are the only ones who want minimum wage raised.
Illegals are not payed minimum wage. They are paid what they are paid.
A higher minimum wage benefits everyone. It gives more money to the working poor to spend on things like clothes and food. Because there is more traffic in stores, those stores need to hire more people at the higher minimum wage. Those people suddenly have more money to spend on things like clothes and food and so forth and so on. None of this happens by giving huge piles of cash to the wealthiest people.
Those low wage earners are not "throw away" people, either. They are making an honest effort to contribute to society. Working to feed their families because nobody but nobody WANTS to be on food stamps. Some of those people have to take two or three of those low wage jobs so they can fight to stay completely off government assistance. Some of those people take low wage jobs because they are out of prison and need something because they don't want to sell drugs and low wage jobs are all they can get.
So many variables.
I think you have it backwards. Of course illegals are not necessarily paid the legal minimum wage. And of course a higher EFFECTIVE (as in real-life, recorded in the labor market) minimum wage benefits everyone. Nobody disputes that. There is nobody, from Donald Trump, the Koch Brothers, to the average Kentucky redneck who wants people at the lower strata of society to earn less. We all want them to earn more. The only people who have an incentive to not want them to earn more are certain politicians and interests groups who thrive on poverty and getting those affected by it mad at someone else - but that's a separate topic.
Simply mandating a higher minimum wage by law is counterproductive. It just encourages cheating (such as hiring illegals) or the employers will seek alternatives, such as automation, or simply forgo hiring and the related activity because it is no longer economically feasible. Mandating a minimum wage squeezes the employer at the bottom end of the scale because it artificially increases his cost but does nothing to allow him to increase his prices to cover the additional cost - because while some of his competitors might be in the same boat and would increase prices, others continue to cheat or are willing to absorb the loss, making a price increase impossible.
The way to increase the effective minimum wage is to do so organically, and by market forces. How do you increase the price of something? You have to decrease the supply relative to demand. If there are 150 applicants for 100 jobs, the market wage will be low. If there are 75 applicants for 100 jobs, the wage will necessarily have to go up.
That is why people have been harping on getting control of illegal immigration for so long - to decrease the supply of low-skill labor and allow minimum wages to float. Look around the world at all the countries who have allowed in large amounts of cheap labor, such as the UK. All of them are suffering from the same thing - the indigenous low-skill population are finding only very low salaries on offer and have a hard time making ends meet - in many cases the offered wage isn't even worth them leaving the house in the morning and they would rather stay at home.
How many times have we heard the argument that "immigrants take the jobs americans don't want to take."? It's a BS argument. If they can hire illegal Jose to work in the tobacco fields at $8 per hour, then that is the prevailing wage and I can understand Americans not really wanting that job in suffient numbers. But what if there were no illegals available to the industry? The farmers would have no choice but to raise their wages. $10, $15, maybe at $20/hour, at some point you will have all sorts of Americans willing to do the toughest jobs, if the wage is good enough. THAT is where you get your increasing minimum wage.
If you want Americans at that end of the wage spectrum to earn more money, that's how you do it. By making the labor they can provide more scarce, and therefore more valuable.
Dictating a minimum wage by law displays a fundamental lack of economic understanding, common to all who have been indoctrinated into Marxist economic principles. Marx said that the value of labor can be calculated mathematically based on various inputs - and socialism takes that further by having the state assign a value to a good or service. By this logic, a minimum wage law makes sense. But as with Marx's theories in general, his LTV is based on bullshit. The actual value of a good or service is nothing more or less than what someone is willing to pay for it at that particular time and place. If the government were to pass a law making the minimum wage $100 per hour, that does not mean the economy can function at that level.
Democrats haven't been this angry since we took away their slaves.