Two-dimensional semiconductor materials with intrinsic magnetism have great application prospects in realizing spintronic devices with low power consumption, small size and high efficiency. Some two-dimensional materials with special lattice structures, such as kagome lattice crystals, are favored by researchers because of their novel properties in magnetism and electronic properties. Recently, a new two-dimensional magnetic semiconductor material Nb<sub>3</sub>Cl<sub>8</sub> monolayer with kagome lattice structure was successfully prepared, which provides a new platform for exploring two-dimensional magnetic semiconductor devices with kagome structure. In this work, we study the electronic structure and magnetic anisotropy of Nb<sub>3</sub>Cl<sub>8</sub> monolayer. We also further construct its <i>p-n</i> junction diode and study its spin transport properties by using density functional theory combined with non-equilibrium Green's function method. The results show that the phonon spectrum of the Nb<sub>3</sub>Cl<sub>8</sub> monolayer has no negative frequency, confirming its dynamic stability. The band gap of the spin-down state (1.157 eV) is significantly larger than that of the spin-up state (0.639 eV). The magnetic moment of the Nb<sub>3</sub>Cl<sub>8</sub> monolayer is 0.997 μ<sub>B</sub>, and its easy magnetization axis is in the plane and along the <i>x</i> axis direction based on its energy of magnetic anisotropy. Nb atoms make the main contribution to the magnetic anisotropy. When the strain is applied, the band gap of the spin-down states will decrease, while the band gap of the spin-up state is monotonously decreased from the negative (compress) to positive (tensile) strain. As the strain variable goes from -6% to 6%, the contribution of Nb atoms to the total magnetic moment gradually increases. Moreover, strain causes the easy magnetization axis of the Nb<sub>3</sub>Cl<sub>8</sub> monolayer to flip vertically from in-plane to out-plane. The designed <i>p-n</i> junction diode nanodevice based on Nb<sub>3</sub>Cl<sub>8</sub> monolayer exhibits an obvious rectification effect. In addition, the current in the spin-up state is larger than that in the spin-down state, exhibiting a spin-polarized transport behavior. Moreover, a negative differential resistance (NDR) phenomenon is also observed, which could be used in the NDR devices. These results demonstrate that the Nb<sub>3</sub>Cl<sub>8</sub> monolayer material has great potential application in the next generation of high-performance spintronic devices, and further experimental verification and exploration of this material and related two-dimensional materials are needed.