A new waveguide-based surface plasmon resonance (SPR) sensor was proposed and investigated by numerical simulation. The sensor consists of a graphene cover layer, a gold (Au) thin film, and a silicon carbide (SiC) waveguide layer on a silicon dioxide/silicon (SiO₂/Si) substrate. The large bandgap energy of SiC allows the sensor to operate in the visible and near-infrared wavelength ranges, which effectively reduces the light absorption in water to improve the sensitivity. The sensor was characterized by comparing the shift of the resonance wavelength peak with change of the refractive index (RI), which mimics the change of analyte concentration in the sensing medium. The study showed that in the RI range of 1.33~1.36, the sensitivity was improved when the graphene layers were increased. With 10 graphene layers, a sensitivity of 2810 nm/RIU (refractive index unit) was achieved, corresponding to a 39.1% improvement in sensitivity compared to the Au/SiC sensor without graphene. These results demonstrate that the graphene/Au/SiC waveguide SPR sensor has a promising use in portable biosensors for chemical and biological sensing applications, such as detection of water contaminations (RI = 1.33~1.34), hepatitis B virus (HBV), and glucose (RI = 1.34~1.35), and plasma and white blood cells (RI = 1.35~1.36) for human health and disease diagnosis.