The photogalvanic effect - a rectified current induced by light irradiation - requires the intrinsic symmetry of the medium to be sufficiently low, which strongly limits candidate materials for this effect. In this paper we explore how in Weyl semimetals the photogalvanic effect can be enabled and controlled by design of the material surface. Specifically, we provide a theory of ballistic linear and circular photogalvanic current in a Weyl semimetal spatially confined to a slab under general and variable surface boundary conditions. The results are applicable to Weyl semimetals with an arbitrary number of Weyl nodes at radiation frequencies small compared to the energy of nonlinear terms in the dispersion at the Fermi level. The confinement-induced response is tightly linked to the configuration of Fermi-arc surface states, specifically the Fermi-arc connectivity and direction of emanation from the Weyl nodes, thus inheriting the same directionality and sensitivity to boundary conditions. As a result, the photogalvanic response of the system becomes much richer than that of an infinite system, and may be tuned via surface manipulations.