Symmetry-protected topological edge modes are one of the most remarkable phenomena in topological physics. Here, we formulate and quantitatively examine the effect of a quantum bath on these topological edge modes. Using the density matrix renormalization group method, we study the ground state of a composite system of spin-1 quantum chain, where the system and the bath degrees of freedom are treated on the same footing. We focus on the dependence of these edge modes on the global and partial symmetries of system-bath coupling and on the features of the quantum bath. It is shown that the time-reversal symmetry (TRS) plays a special role for an open quantum system, where an emergent partial TRS breaking will result in a TRS-protected topological mode diffusing from the system edge into the bath, thus make it useless for quantum computation.