In this work we theoretically investigate the responsivity/noise equivalent power (NEP) trade-off in graphene/semiconductor Schottky photodetectors (PDs) operating in the near-infrared regime and working at room temperature. Our analysis shows as the responsivity/NEP ratio is strongly dependent on the Schottky barrier height (SBH) of the junction and we derive a closed analytical formula for maximizing it. In addition, we theoretically discuss as the SBH is linked to the bias applied to the junction in order to show how these devices could be optimized in practice for different semiconductors. We discover that graphene/n-silicon (Si) Schottky PDs could be optimized at 1550nm showing a responsivity and NEP of 133mA/W and 500fW/Hz, respectively, by a low reverse bias of only 0.66V. Moreover, we show that graphene/n-germanium (Ge) Schottky PDs optimized in term of responsivity/NEP ratio could be employed at 2000nm with a responsivity and NEP of 233mA/W and 31pW/Hz, respectively. We believe that our insights are of great importance in the field of silicon photonics for the realization of Si-based PDs to be employed in power monitoring, lab-on-chip and environment monitoring applications.