We propose a biosensor by exploiting localized plasmons in graphene and biomolecule adsorption on it. Numerical simulations demonstrate that the sensitivity of such a device can achieve a high value of up to 1697 nm/RIU (refractive index unit) when the wavelength shift at the plasmon resonance is detected. The transparent substrate supporting graphene can be chosen potentially from a wide range of materials including insulators, semiconductors, polymers, and gels. The plasmon resonance wavelength can be tuned with electrostatic doping and/or structure modulation of graphene. Furthermore, the device works in a wide angle range of incident light since the transverse magnetic (TM) polarization is independent of incident angles.