The solution for having minority representation is to use some sort of proportional representation with a multiple-winner system. But, in elections that can only have one winner (i.e. for president, governor, mayor, etc...) there's no way around the risk that a majority of people have bad opinions. That's not a problem with the voting system, but with the electorate.
Or you do what parliamentary systems do and have the executive position be chosen by the elected representatives, and try to set it up so that minority representation still has to be taken into account to "form a government" as the Brits say.
