Employment is a transaction, where the employer is willing to pay up to X and the employee willing to work for no less than Y. For jobs where the minimum wage is greater than X it kills the job, but when it is less than X but more than Y it can force more of the surplus from the employment transaction to go to the employee. Obviously it's hard to set the right minimum wage for all jobs in a large economy.

Also, it's my understanding that people working above but near the minimum wage usually get raises when the minimum wage goes up; the minimum wage communicates something about what a worker is supposed to be worth, and telling your employees they are the bottom of the barrel is probably bad for morale, and thus, productivity. So the minimum wage can affect wages even if they aren't at the minimum wage.