In the US, the specifics depend on the state. It’s often not “primary enforcement”, which means you can’t be pulled over specifically for that, but it can be added on when pulled over for something else. In some states, yes, the driver can be held responsible for passengers not having seatbelts on. It may also matter if the passenger is a minor.

Primary enforcement of seatbelt laws tends to make it more likely for black people to be pulled over. It’s amazing how many good ideas get ruined by racism once you dig into the details.