Metasurfaces based on quasi-bound states in the continuum (quasi-BICs) constitute an emerging toolkit in nanophotonic sensing as they sustain high quality factor resonances and substantial near-field enhancements. It is demonstrated that silicon metasurfaces composed of crescent shaped meta-atoms provide tailored light-matter interaction controlled by the crescent geometry. Significantly, this metasurface not only exhibits a fundamental quasi-BIC resonance, but also supports a higher-order resonance with tunable electromagnetic field enhancement and advantageous properties for sensing. The higher-order resonance shows twice the sensitivity of the fundamental one for bulk refractive index sensing. It is further demonstrated that both the fundamental and higher-order resonances can be exploited for sensing ultrathin layers of biomolecules in air and buffer solutions. Specifically, when measuring in buffer solution, the figure of merit of the sensor, defined as the change in the spectral position of the resonance normalized to its full width at half maximum, is a factor of 2.5 larger for the higher-order resonance when compared to the fundamental one. Due to its high sensitivity and potential for straightforward microfluidic integration, the silicon crescent metasurface is ideally suited for real-time and in situ biosensing, enabling compact sensing devices for a wide range of diagnostic applications.