We report a detailed study of the temperature and magnetic-field dependence of the spin susceptibility for a single crystal of La(1.875)Ba(0.125)CuO(4). From a quantitative analysis, we find that the temperature-dependent anisotropy of the susceptibility, observed in both the paramagnetic and stripe-ordered phases, directly indicates that localized Cu moments dominate the magnetic response. A field-induced spin-flop transition provides further corroboration for the role of local moments. Contrary to previous analyses of data from polycrystalline samples, we find that a commonly-assumed isotropic and temperature-independent contribution from free carriers, if present, must be quite small. Our conclusion is strengthened by extending the quantitative analysis to include crystals of La(2-x)Ba(x)CuO(4) with x=0.095 and 0.155. On the basis of our results, we present a revised interpretation of the temperature and doping dependence of the spin susceptibility in La(2-x)(Sr,Ba)(x)CuO(4).