When a symmetry is spontaneously broken what happens to the corresponding Noether's charge? Since the action never breaks the symmetry only the vacuum does do we expect the charge to remain conserved?
The conservation of Noether's charge relies on the assumption that the 3-current vanish at the boundary of spacetime which in turn requires the field $\phi(x)$ itself to vanish at spatial infinity. In case of spontaneously broken symmetry, although the action does not break the symmetry, the field at infinity does not vanish but goes to a constant value corresponding to one of the minima of the classical potential. This is the same reason why odd correlation functions can be nonzero in a spontaneously broken $\phi^4$-theory even though the action itself respects the $\phi\to-\phi$ symmetry.