The exotic topological surface states of Dirac or Weyl semimetals, namely Fermi arcs, are predicted to be spin polarized, while their spin polarization nature is still not revealed by transport measurements. Here, we report the spin-polarized transport in a Dirac semimetal Cd_{3}As_{2} nanowire employing the ferromagnetic electrodes for spin detection. The spin-up and spin-down states can be changed by reversing the current polarity, showing the spin-momentum locking property. Moreover, the nonlocal measurements show a high fidelity of the spin signals, indicating the topological protection nature of the spin transport. As tuning the Fermi level away from the Dirac point by gate voltages, the spin signals gradually decrease and finally are turned off, which is consistent with the fact that the Fermi arc surface state has the maximum ratio near the Dirac point and disappears above the Lifshitz transition point. Our results should be valuable for revealing the transport properties of the spin-polarized Fermi arc surface states in topological semimetals.