Early development of spinodal instabilities and density correlation functions in asymmetric nuclear matter are investigated in the stochastic extension of the Walecka-type relativistic mean field including coupling with rho meson. Calculations are performed under typical conditions encountered in heavy-ion collisions and in the crusts of neutron stars. In general, growth of instabilities occur relatively slower for increasing charge asymmetry of matter. At higher densities around rho = 0.4 rho(0) fluctuations grow relatively faster in the quantal description than those found in the semi-classical limit. Typical sizes of early condensation regions extracted from density correlation functions are consistent with those found from dispersion relations of the unstable collective modes.