Two of the most attractive spectral windows for spectropolarimetric investigations of the physical properties of the plasma structures in the solar chromosphere and corona are the ones provided by the spectral lines of the He I 10830 and 5876 Å (or D3) multiplets, whose polarization signals are sensitive to the Hanle and Zeeman effects. However, in order to be able to carry out reliable diagnostics, it is crucial to have a good physical understanding of the sensitivity of the observed spectral line radiation to the various competing driving mechanisms. Here we report a series of off-the-limb non-LTE calculations of the He I D3 and 10830 Å emission profiles, focusing our investigation on their sensitivity to the EUV coronal irradiation and the model atmosphere used in the calculations. We show in particular that the intensity ratio of the blue to the red components in the emission profiles of the He I 10830 Å multiplet turns out to be a good candidate as a diagnostic tool for the coronal irradiance. Measurements of this observable as a function of the distance to the limb and its confrontation with radiative transfer modeling might give us valuable information on the physical properties of the solar atmosphere and on the amount of EUV radiation at relevant wavelengths penetrating the chromosphere from above.