Graphene photodetectors have drawn a lot of interest due to their superior properties. For entering real applications, the existing open-face graphene photodetectors need to be properly protected to obtain long term stability. Here we demonstrate a facile method to obtain air-stable graphene photodetectors by directly depositing Si3N4 layer on the surface of buried-gate graphene field-effect transistors (GFETs). The electrical and photoelectrical properties of the Si3N4-encapsulated GFETs were carefully investigated and compared with that of the reference open-face GFETs. A photoresponsivity of 6.3 mA/W was obtained under a gate bias of 6 V, which was over 4 times higher than the photoresponsivities of typical back-gated GFET photodetectors under gate biases up to 80 V. Furthermore, the Si3N4-encapsulated GFET photodetectors demonstrated good stability for several months.