The structural, dynamical and dielectric properties of the cubic phase of perovskite barium stannate BaSnO3, a potential candidate as protonic conductor for solid oxide fuel cells, have been investigated by the means of first-principles density functional calculations, and the structural and electrical properties have been explored at low temperature. From density functional perturbative calculations, the phonon modes, the Born effective charges and the dielectric tensor are derived and analyzed, at zero pressure. The phonon band-structure of the cubic phase does not exhibit unstable modes, in good agreement with x-ray diffraction, which shows that BaSnO3 remains perfectly cubic down to 10 K. The dielectric response in BaSnO3 as measured and calculated is lower than in titanate and zirconate perovskites.