We fabricate dual-gated electric double layer (EDL) field effect transistors based on InAs nanowires gated with an ionic liquid, and we perform electrical transport measurements in the temperature range from room temperature to 4.2 K. By adjusting the spatial distribution of ions inside the ionic liquid employed as gate dielectric, we electrostatically induce doping in the nanostructures under analysis. We extract low-temperature carrier concentration and mobility in very different doping regimes from the analysis of current-voltage characteristics and transconductances measured exploiting global back-gating. In the liquid gate voltage interval from -2 to 2 V, carrier concentration can be enhanced up to two orders of magnitude. Meanwhile, the effect of the ionic accumulation on the nanowire surface turns out to be detrimental to the electron mobility of the semiconductor nanostructure: the electron mobility is quenched irrespectively to the sign of the accumulated ionic species. The reported results shine light on the effective impact on crucial transport parameters of EDL gating in semiconductor nanodevices and they should be considered when designing experiments in which electrostatic doping of semiconductor nanostructures via electrolyte gating is involved.