The argument for harm should be re-framed in terms of minimum wages devaluing the buying power of a dollar, of raises to those wages as a tax upon the middle-class (who's wages presumably do not increase accordingly), but crucially not the rent taking (wealthy) class.
Rather than focusing just on dollars and an hourly wage, efforts must be made to correct the market for life essentials so that everyone is able to invest in themselves and the future. Taxes should be structured to prevent the wealthy from abusing the poor. There are other market distortions as well, such as artificially limited housing and many other forms of ladders being kicked out from behind of those who already have their place in life, and don't care for those whom are still seeking a place of their own to belong.