We present measurements of [SIII]$\lambda\lambda$9069,9531 for a sample of $z\sim1.5$ star-forming galaxies, the first sample with measurements of these lines at z>0.1. We employ the line ratio S$_{32}$$\equiv$[SIII]$\lambda\lambda$9069,9531/[SII]$\lambda\lambda$6716,6731 as a novel probe of evolving ISM conditions. Since this ratio includes the low-ionization line [SII], it is crucial that the effects of diffuse ionized gas (DIG) on emission-line ratios be accounted for in $z\sim0$ integrated galaxy spectra, or else that comparisons be made to samples of local HII regions in which DIG emission is not present. We find that S$_{32}$ decreases with increasing stellar mass at both $z\sim1.5$ and $z\sim0$, but that the dependence is weak suggesting S$_{32}$ has a very shallow anticorrelation with metallicity, in contrast with O$_{32}$ that displays a strong metallicity dependence. As a result, S$_{32}$ only mildly evolves with redshift at fixed stellar mass. The $z\sim1.5$ sample is systematicallty offset towards lower S$_{32}$ and higher [SII]/H$\alpha$ at fixed [OIII]/H$\beta$ relative to $z=0$ HII regions. By comparing to photoionization model grids, we find that such trends can be explained by a scenario in which the ionizing spectrum is harder at fixed O/H with increasing redshift, but are inconsistent with an increase in ionization parameter at fixed O/H. This analysis demonstrates the advantages of expanding beyond the strongest rest-optical lines for evolutionary studies, and the particular utility of [SIII] for characterizing evolving ISM conditions and stellar compositions. These measurements provide a basis for estimating [SIII] line strengths for high-redshift galaxies, a line that the James Webb Space Telescope will measure out to z~5.5.