We propose and analyze surface-plasmon-driven electron spin currents in a thin metallic film. The electron gas in the metal follows the transversely rotating electric fields of the surface plasmons (SPs), which leads to a static magnetization gradient. We consider herein SPs in a thin-film insulator-metal-insulator structure and solve the spin diffusion equation in the presence of a magnetization gradient. The results reveal that the SPs at the metal interfaces generate spin currents in the metallic film. For thinner film, the SPs become strongly hybridized, which increases the magnetization gradient and enhances the spin current. We also discuss how the spin current depends on SP wavelength and the spin-diffusion length of the metal. The polarization of the spin current can be controlled by tuning the wavelength of the SPs and/or the spin diffusion length.