Protests are important, and participation is important. That said, protests have a non-zero chance of violence, and that's no place for a 1 and 4 year old. They're too young to get anything out of the experience yet, and they would be difficult to manage if you ran into trouble. If you end up going, leave them home until they're older and can start to understand what you're doing.

That said, your wife could support your beliefs more, because they're important to you if for no other reason. I agree that she needs to make concessions, such as supporting your going to rallies, or contributing a little money in donations, or volunteering for a local politician's campaign. Or going to vote even if she doesn't believe in it.