Does hiding our nationality keep us safe?

I wasn't there when it happened, but my dad was taking my mom to the memorial hospital in Long Beach to get a check up on her diabetes and get a refill on her medicine when an old man told him to go back to Korea. Security escorted the man away but it's shocking to hear it still that someone would say that in public. After they told my husband and I about it, my mother warned me, "Don't dye your hair back to black for a while and to just leave it blonde so that if and when we go out no one will know right away that you’re Asian.” Like how messed up is that?? That's how some of us are thinking nowadays, that we would have to hide our nationality to just stay safe.