The relationship between the emission of coronal lines (CLs) and nuclear activity in 36 Type 1 and 18 Type 2 active galactic nuclei (AGNs) is analyzed, for the first time, based on near-infrared (0.8-2.4 μm) spectra. The eight CLs studied, of Si, S, Fe, Al, and Ca elements and corresponding to ionization potentials (IPs) in the range 125-450 eV, are detected (3σ) in 67% (36 AGNs) of the sample. Our analysis shows that the four most frequent CLs [Si VI] 1.963 μm, [S VIII] 0.9913 μm, [S IX] 1.252 μm, and [Si X] 1.430 μm display a narrow range in luminosity, with most lines located in the interval log L 39-40 erg s-1. We found that the non-detection is largely associated with either loss of spatial resolution or increasing object distance: CLs are essentially nuclear and easily lose contrast in the continuum stellar light for nearby sources or get diluted by the strong AGN continuum as the redshift increases. Yet, there are AGNs where the lack of coronal emission, i.e., lines with IP >= 100 eV, may be genuine. The absence of these lines reflects a non-standard AGN ionizing continuum, namely, a very hard spectrum lacking photons below a few Kev. The analysis of the line profiles points out a trend of increasing FWHM with increasing IPs up to energies around 300 eV, where a maximum in the FWHM is reached. For higher IP lines, the FWHM remains nearly constant or decreases with increasing IPs. We ascribe this effect to an increasing density environment as we approach the innermost regions of these AGNs, where densities above the critical density of the CLs with IPs larger than 300 eV are reached. This sets a strict range limit for the density in the boundary region between the narrow and the broad region of 108-109 cm-3. A relationship between the luminosity of the CLs and that of the soft and hard X-ray emission and the soft X-ray photon index is observed: the coronal emission becomes stronger with both increasing X-ray emission (soft and hard) and steeper X-ray photon index, i.e., softer X-ray spectra. Thus, photoionization appears as the dominant excitation mechanism. These trends hold when considering Type 1 sources only; they get weaker or vanish when including Type 2 sources, very likely because the X-ray emission measured in the latter is not the intrinsic ionizing continuum.