We investigated the electrical characteristics of molecular electronic devices consisting of benzenedithiolate self-assembled monolayers and a graphene electrode. We used the multilayer graphene electrode as a protective interlayer to prevent filamentary path formation during the evaporation of the top electrode in the vertical metal-molecule-metal junction structure. The devices were fabricated both on a rigid SiO2/Si substrate and on a flexible poly(ethylene terephthalate) substrate. Using these devices, we investigated the basic charge transport characteristics of benzenedithiolate molecular junctions in length- and temperature-dependent analyses. Additionally, the reliability of the electrical characteristics of the flexible benzenedithiolate molecular devices was investigated under various mechanical bending conditions, such as different bending radii, repeated bending cycles, and a retention test under bending. We also observed the inelastic electron tunneling spectra of our fabricated graphene-electrode molecular devices. Based on the results, we verified that benzenedithiolate molecules participate in charge transport, serving as an active tunneling barrier in solid-state graphene-electrode molecular junctions.