We study asymptotic stability of an infinite dimensional system that switches between a finite set of modes. Each mode is governed by a system of one-dimensional, linear, hyperbolic partial differential equations on a bounded space interval. The switching system is fairly general in that the space dependent system matrix functions as well as the boundary conditions may switch in time. For the case in which the switching occurs between subsystems in canonical diagonal form, we provide two sets of sufficient conditions for asymptotic stability under arbitrary switching signals. These results are direct generalizations of the corresponding results for the unswitched case. Furthermore, we provide an explicit dwell-time bound on the switching signals that guarantee asymptotic stability of the switched system under the assumption that each of the subsystems are stable. Our results of stability under arbitrary switching generalize to the case where switching occurs between non-diagonal hyperbolic systems that are diagonalizable using a common transformation. For the case where no such transformation exists, we prove existence of a dwell-time bound on the switching signals such that asymptotic stability is guaranteed. To motivate our study, we discuss a potential application to stability of water flow in one-dimensional open channels governed by linearized Saint-Venant equations.