I don't think they care what their family thinks of them .... I think they care about how angry democratically minded citizens see them as unidentified fascists hiding behind masks

If what you are doing is legal, legitimate and moral ... then you never have to mask your name or hide your face

If what you are doing is illegal, immoral and highly dangerous to everyone including you and your family and you know it .. then you are more apt to want to hide your identity and avoid having to deal with the repercussions of what you know is wrong