The PHENIX Collaboration has measured the ratio of the yields of psi(2S) to psi(1S) mesons produced in p + p, p + Al, p + Au, and He-3+Au collisions at root S-NN = 200 GeV over the forward and backward rapidity intervals 1.2 < | y | < 2.2. We find that the ratio in p + p collisions is consistent with measurements at other collision energies. In collisions with nuclei, we find that in the forward (p-going or He-3-going) direction, the relative yield of psi(2S) mesons to psi(1S) mesons is consistent with the value measured in p + p collisions. However, in the backward (nucleus-going) direction, the psi(2S) meson is preferentially suppressed by a factor of similar to 2. This suppression is attributed in some models to the breakup of the weakly bound psi(2S) meson through final-state interactions with comoving particles, which have a higher density in the nucleus-going direction. These breakup effects may compete with color screening in a deconfined quark-gluon plasma to produce sequential suppression of excited quarkonia states.