A single-mode optical fiber sensor uses surface plasmon resonance (SPR) with a bimetallic silver–copper (Ag–Cu) coating compared to a single layer of Ag and Cu itself. Bimetallic Ag–Cu sensors are constructed by simple fabrication on a side-polished optical fiber, followed by an electron beam evaporation of Ag and Cu films. For this investigation, the thickness of the single Ag layer was set to 30 nm and the single Cu layer was set to 30 nm; whereas for the bimetallic combined Ag–Cu layer the thickness of Ag was 7 nm and Cu 23 nm. The sensor performance was analyzed and compared experimentally and numerically using the COMSOL Multiphysics. A white light source was used with a broad optical bandwidth to provide a range of wavelengths to the optical fiber. The characteristics of the thin layers of Ag, Cu, and Ag–Cu as alcohol sensors were evaluated. We found that Cu was the most sensitive metallic layer compared to the Ag and the bimetallic Ag–Cu layers. For a 100% alcohol concentration, Cu showed a sensitivity of 425 nm/RIU followed by the bimetallic Ag–Cu layer with 108.33 nm/RIU, whereas the Ag layer was not detected. Interestingly, sensitivity reached saturation beyond the 20 nm thick layer of Ag. This shows that the Cu and the bimetallic Ag–Cu layers are suitable for an alcohol-based optical sensor.