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We introduce a classification of symmetry-protected topological phases applicable to stationary states of open systems based on a generalization of the many-body polarization. The polarization can be used to probe topological properties of noninteracting and interacting closed and open systems and remains a meaningful quantity even in the presence of moderate particle-number fluctuations. As examples, we discuss two open-system versions of a topological Thouless pump in the steady state of one-dimensional lattices driven by Markovian reservoirs. In an analogous unitary system, the Su-Shrieffer-Heeger model, symmetries enforce a quantization of the geometric Zak phase, which acts as a topological invariant. Introducing a further degree of freedom, a nontrivial winding of the phase can be observed upon cyclic variations of parameters. Associated with this is a quantization or, respectively, a winding of the polarization corresponding to a quantized transport (Thouless pump). We here show that in the open system, where the Zak phase loses its meaning, the same symmetries enforce a quantization and more generally a winding of the generalized many-body polarization. These features are shown to be robust against Hamiltonian perturbations as well as homogeneous dephasing and particle losses. [ABSTRACT FROM AUTHOR]