Double-spin asymmetries of semiinclusive cross sections for the production of identified pions and kaons have been measured in deep inelastic scattering of polarized positrons on a polarized deuterium target. Five helicity distributions including those for three sea quark flavors were extracted from these data together with reanalyzed previous data for identified pions from a hydrogen target. These distributions are consistent with zero for all three sea flavors. A recently predicted flavor asymmetry in the polarization of the light quark sea appears to be disfavored by the data.