How does one keep to their convictions even though it puts their family in danger?

Well, Albert Camus said: if I had to choose between justice and my mother, I would choose my mother.

They feel it is the morally right thing to do. Look at all the rescuers of the Holocaust their families could have been killed. Some of them and their families were killed for trying to save the Jews. The ones who lived all pretty much state they did it because it was the right thing to do.

I think you have to weigh the risk of danger against the risk of teaching your family that values are not actually important. We all want our children to grow up to be good people. It's the right thing to teach them that some things are bigger than ourselves, and that some things are worth dying for.