I'm surprised that there's such an expectation of this in the United States - I remember being vaguely appalled when I went to write my AP exams here in Canada and they wanted a SIN! Do you guys spread it out like butter on toast? Seems the States uses it practically everywhere, whereas in Canada it's actually considered a minor offence to throw it out willy-nilly.
As far as the government goes, the only way to stop the hassle from your SSN being stolen, is to die. It wont stop the fraud or the people harassing you or your family, the only change will be that your not longer alive to feel hassle personally. When any citizen has a serious problem, like the police raiding their house because someone else used their identity, and the government is directly complicit in the situation, since they force the SSN on you while at the same time making it so pathetically easy for someone else to steal and use it, and the governments only proposed solution that would completely resolve the issue for you, is for you to die, then there is a serious problem with said government.