We investigate the stability of a domain wall in the striped phase with simultaneous charge and spin ordering in doped planar antiferromagnets. The line defects gain magnetic energy but the localization of holes costs kinetic energy. A Hartree-Fock approximation is used to estimate the energy cost of a static kink in the domain wall. Such kinks can gain kinetic energy by delocalizing along the line. We find that for values of t>or approximately=J this energy gain surpasses the cost in magnetic energy so that a quantum melting of the domain walls through the proliferation of kinks is predicted for this parameter regime.