One of the most useful spectral windows for spectropolarimetric investigations of the solar chromosphere is the one provided by the spectral lines of the He I 10830 Å multiplet, whose polarization signals are sensitive to the Hanle and Zeeman effects. However, in order to be able to carry out reliable diagnostics of the dynamic and magnetic properties of the solar outer atmosphere it is crucial to have a good physical understanding of the sensitivity of the observed spectral line radiation to the various competing triggering mechanisms. Here we report a series of on-disk and off-the-limb non-LTE calculations of the 10830 Å absorption and emission profiles, focusing our investigation on their sensitivity to the EUV coronal irradiation and the model atmosphere used in the calculations. We show in what respects the on-disk case sensitivity of the polarization signals induced by the Zeeman effect to the EUV coronal irradiance, and investigate whether or not inversions based on the Milne-Eddington model are reliable. Concerning the off-the-limb case we demonstrate that the intensity ratio of the blue to the red components of the He I 10830 Å multiplet is a sensitive function of the amount of EUV coronal illumination. Therefore, 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 penetrating the chromosphere from above.