We investigate the Relative Intensity Noise (RIN) properties of a multi-Stokes Brillouin fiber ring laser. We experimentally analyse intensity noise of each Stokes waves and study the noise dynamics of the cascaded Brillouin scattering process. We observe up to 20 dB/Hz intensity noise reduction compared to that of the RIN input pump laser. We examine the impact of the fiber ring quality factor on the laser RIN features such as amplitude reduction and relaxation frequency. A numerical model based on a set of coupled-mode equations replicate the experimental observations; confirming the class B like behavior of a multi-Stokes Brillouin laser. Our study enables to determine the optimal parameter values to operate the multi-Stokes laser in the low noise regime.