We present a reliable methodology to perform electron transport measurements at the nanoscale on supported biomembranes by conductive atomic force microscopy (C-AFM). It allows measurement of both (a) non-destructive conductive maps and (b) force controlled current-voltage characteristics in wide voltage bias range in a reproducible way. Tests experiments were performed on purple membrane monolayers, a two-dimensional (2D) crystal lattice of the transmembrane protein bacteriorhodopsin. Non-destructive conductive images show uniform conductivity of the membrane with isolated nanometric conduction defects. Current-voltage characteristics under different compression conditions show non-resonant tunneling electron transport properties, with two different conduction regimes as a function of the applied bias, in excellent agreement with theoretical predictions. This methodology opens the possibility for a detailed study of electron transport properties of supported biological membranes, and of soft materials in general.