We propose a model for conventional superconductors in contact with a ferromagnetic (or polarized paramagnetic) insulator. The model is defined by a boundary condition on the quasiclassical Green's function for the superconductor at the interface between the metal and insulating magnet. The specific boundary condition we use describes the interaction of the electrons, which tunnel into the insulating barrier, with the average exchange field of the local moments. Solutions to the quasiclassical equations and boundary condition are obtained for thin superconducting films. We obtain results for pair-breaking effects of a magnetic boundary on the transition temperature and gap of thin superconducting films. Of particular interest is the Zeeman effect in the quasiparticle density of states (DOS), which exhibits a splitting of the form 2μe(H + B*) in an external field H. The excess splitting B* is interpreted here as an internal field in the superconductor resulting from quasiparticle tunneling into the magnetic insulator, and subsequent reflection into the superconducting film. We compare our model of the tunneling DOS with the recent measurements by Tedrow et al. on EuO/Al/Al2O3 tunnel junctions.