Monte Carlo simulations are performed to determine the anisotropy of the electric polarizability of a model DNA fragment in aqueous salt solution. By taking into consideration the participation of coions in the electroneutrality condition, at every simulation step, we obtain a list of counterions constituting the net charge arranged in increasing order of their distance from the DNA and calculate the contribution to the dipole moment from the first n counterions in the list. We define a partial polarizability tensor due to these n counterions to understand the origin of the polarizability in close relation to the solution structure. The ionic distributions are described by the counterion condensation theory. Characteristic features of the electric properties of polyelectrolytes are reproduced. The anisotropy of the electric polarizability Delta alpha of DNA decreases with the addition of salt, yielding values comparable to experiment. The effect of electrophoretic motion of the polyion is examined by estimating its upper limit.