The potential therapeutic applications of mesenchymal stem/stromal cells (MSCs) and biomaterials have attracted a great amount of interest in the field of biomedical engineering. MSCs are multipotent adult stem cells characterized as cells with specific features, e.g., high differentiation potential, low immunogenicity, immunomodulatory properties, and efficient in vitro expansion ability. Human umbilical cord Wharton’s jelly-derived MSCs (hUC-MSCs) are a new, important cell type that may be used for therapeutic purposes, i.e., for autologous and allogeneic transplantations. To improve the therapeutic efficiency of hUC-MSCs, novel biomaterials have been considered for use as scaffolds dedicated to the propagation and differentiation of these cells. Nowadays, some of the most promising materials for tissue engineering include graphene and its derivatives such as graphene oxide (GO) and reduced graphene oxide (rGO). Due to their physicochemical properties, they can be easily modified with biomolecules, which enable their interaction with different types of cells, including MSCs. In this study, we demonstrate the impact of graphene-based substrates (GO, rGO) on the biological properties of hUC-MSCs. The size of the GO flakes and the reduction level of GO have been considered as important factors determining the most favorable surface for hUC-MSCs growth. The obtained results revealed that GO and rGO are suitable scaffolds for hUC-MSCs. hUC-MSCs cultured on: (i) a thin layer of GO and (ii) an rGO surface with a low reduction level demonstrated a viability and proliferation rate comparable to those estimated under standard culture conditions. Interestingly, cell culture on a highly reduced GO substrate resulted in a decreased hUC-MSCs proliferation rate and induced cell apoptosis. Moreover, our analysis demonstrated that hUC-MSCs cultured on all the tested GO and rGO scaffolds showed no alterations of their typical mesenchymal phenotype, regardless of the reduction level and size of the GO flakes. Thus, GO scaffolds and rGO scaffolds with a low reduction level exhibit potential applicability as novel, safe, and biocompatible materials for utilization in regenerative medicine.