Even a few Republican politicians (ahem, Ben Carson) struggle with the basic economic concept that when the price of a good is raised, less of it is used.
The minimum wage is predominantly paid to second income earners who need the work to support a family and perhaps most importantly, youths and the unskilled. Putting a price floor inevitably will result in many of these individuals being eliminated from and priced out of the market. Particularly in the case of the young and the unskilled, this will relegate them to a lifetime of earning less than they could have, as that first job can serve as the crucial first rung up the ladder that puts them on a path to rising skills and rising incomes. With no job to be had in the first place due to the market artificially pricing them out, they will be much more likely to struggle and develop dependence on family or even worse, the government (or more correctly, us thr taxpayers). The later and individual can get their first foot on the ladder, the slower they will climb up it.
I am thankful that as a high school and college age kid that I could find employment at a grocer bagging and carrying out groceries, at a feed mill loading feed sacks and fencing material, and at a plumbing company running parts and digging ditches (cue the Caddyshack quote on the world needing ditchdiggers too). These were critical discipline building occupations for me as well as serving as critical funds helping me pay for my college education. The grocery store I worked at had slim 4 percent margins. A too high of a minimum wage would have brought about the necessity to push customers to more self service kiosks for checkout and forcing them to carry out their own groceries, and thus the likely elimination of my job.
Simply because a policy feels good and tugs on the emotional heartstrings of “helping the little guy” does not mean that an economically fallacious policy will actually bring that about. In this case, I would argue that it is perniciously counterproductive to the end goal.