Polarization refers to the orientation of the wave oscillation which is a fundamental property of wave. It has been used widely to encode information in photonics and phononics. In magnonics, spin wave also has been used for transmitting and processing information. However, exploiting the spin wave polarization to design devices has not been achieved yet in ferromagnets as only the right-handed polarized spin waves can be accommodated in ferromagnets. Our eariler study suggests that the left-handed polarized spin waves can be introduced into ferromagnets by appling a spin-polarized electric current, thus making it possible to design spin wave devices with polarization encoding. But the critical current needed to induce left-handed polarized spin wave in a uniformly magnetized ferromagnet is too high to be realized experimentally. Magnetic domain wall can serve as spin wave guide, and the cutoff frequency of spin wave in a domain wall approaches zero. In this work, the dispersion relationship and propagation characteristics of spin wave in a Bloch domain wall are studied based on the Landau-Lifshitz equation in the presence of a spin-polarized electrical current. It is found that the stable left-handed spin wave can be generated in the domain wall with only a small current density. Micromagnetic simulations confirm the theoretical analysis results. In addition, due to the different excitation efficiencies and spin transfer torque induced propagating nonreciprocity of left- and right-handed polarized spin wave, it is possible to excite selectively the left- and right-handed polarized spin wave, as well as nearly linearly polarized spin waves. This study provides a practical and feasible solution for designing spin wave devices based on the polarization coding technique.