We present and analyze a protocol for driven-dissipatively preparing and stabilizing a manifold of quantum many-body entangled states with symmetry-protected topological order. Specifically, we consider the experimental platform consisting of superconducting transmon circuits and linear microwave resonators. We perform theoretical modeling of this platform via pulse-level simulations based on physical features of real devices. In our protocol, transmon qutrits are mapped onto spin-1 systems. The qutrits' sharing of nearest-neighbor dispersive coupling to a dissipative microwave resonator enables elimination of state population in the
S
total
=
2
subspace for each adjacent pair, and thus, the stabilization of the many-body system into the Affleck, Kennedy, Lieb, and Tasaki state up to the edge mode configuration. We also analyze the performance of our protocol as the system size scales up to four qutrits, in terms of its fidelity as well as the stabilization time. Our work shows the capacity of driven-dissipative superconducting cQED systems to host robust and self-corrected quantum many-body states that are topologically nontrivial.