In general, minimum wage affects the labor market negatively. At some level, it will increase the unemployment rate. That may happen either through direct disemployment as employers struggle to meet the cost, or through the freezing or insufficient job creation growth as the labor force increase. Whatever it is, I believe the relationship between minimum wage and unemployment rate is relatively well-publicized and many who are serious about the issue do know of the relationship. The lay proponents of minimum wage still promote their policy but they do know that relationship is a wall to scale.
Here is another and it is the distribution effect across age.
Consider two workers of the same skills. Worker A is 25 years old. Worker B is 50 years old.
Both qualify for minimum wage.
If an employer had to choose between the two for a low-skilled job, which would the employer employ?
Without hesitation, I would take the younger one if I was the employer.
Between a 25 years old and at 50 years old, it is very likely that the 25 years old will be the preferred choice of anyone with profit-motive. He is young and that means he has better health than his older counterpart in general. There are other factors of course like attitude and initiative (if the particular person in his 20s is a damn punk and the 50 years old person is a nice old lady, I will employ the lady) but there are many reasons to think that an employer can squeeze more productivity out of the young worker than out of the older worker for a given wage, on average.
For those who know their economic jargon, then that means the younger worker offers better marginal product than the older worker will on average. In simpler terms, the younger worker offers greater productivity than the older worker.
How about experience? Surely experience works in favor of the older workers, right? Remember however that low-skilled jobs require little training. The kind of jobs requires no or little experience. That effectively discounts experience as a consideration.
When one pays a person according to his or her productivity without any restriction on compensation, then one can employ anybody up to any number until your last marginal product of labor is no longer positive. Note the causality: your productivity determines your wage. The first determination is your productivity and your wage is a function of your productivity.
Under minimum wage, the wage is the first determination and your productivity now is a function of your wages. Here, wage is the first determination because an employ know his cost and he will want to find workers with the productivity that matches the cost that is minimum wage. This immediately limit the kind of workers that the employer will employ.
Now, go back to the productivity of the young and the older workers mentioned in the beginning. Older workers will have lower productivity to younger workers. That is an immediate disadvantage in terms of employability in the age of minimum wage.
I think this point is important because a lot of younger workers do not really need a job. Many are out of school and are merely looking for extra pocket money to have some fun. These young workers will qualify for minimum wage. They do not need the jobs. The jobs are merely summer job so-to-speak, not necessarily part-time too.
Compare this to older workers who qualify for minimum wage. This type of older workers will likely need the minimum wage jobs more than the younger workers. They are in it to survive.
Controlling for everything else, minimum wage can hurt the workers that, arguably, the policy of minimum wage is supposed to help. Yet, the policy hurt those that it is set out to help.