We theoretically investigate the spin current for a parabolically confined semiconductor heterojunction quantum wire with weak Rashba spin-orbit coupling by means of the perturbation method. By analytical calculation, it is found that only two components off spin current density is non-zero in the equilibrium case. Numerical examples have demonstrated that the spin current of electron transverse motion is 10-3 times that off electron longitudinal motion. However, the former one is much more sensitive to the strength of Rashba spin-orbit coupling. These results may suggest an approach to the spin storage device and to the measurement of spin current through its induced electric field.