Using observations from the MOSFIRE Deep Evolution Field (MOSDEF) survey, we investigate the physical conditions of star-forming regions in \(z\sim2.3\) galaxies, specifically the electron density and ionization state. From measurements of the [O II]\(\lambda\lambda\)3726,3729 and [S II]\(\lambda\lambda\)6716,6731 doublets, we find a median electron density of \(\sim250\) cm\(^{-3}\) at \(z\sim2.3\), an increase of an order of magnitude compared to measurements of galaxies at \(z\sim0\). While \(z\sim2.3\) galaxies are offset towards significantly higher O\(_{32}\) values relative to local galaxies at fixed stellar mass, we find that the high-redshift sample follows a similar distribution to the low-metallicity tail of the local distribution in the O\(_{32}\) vs. R\(_{23}\) and O3N2 diagrams. Based on these results, we propose that \(z\sim2.3\) star-forming galaxies have the same ionization parameter as local galaxies at fixed metallicity. In combination with simple photoionization models, the position of local and \(z\sim2.3\) galaxies in excitation diagrams suggests that there is no significant change in the hardness of the ionizing spectrum at fixed metallicity from \(z\sim0\) to \(z\sim2.3\). We find that \(z\sim2.3\) galaxies show no offset compared to low-metallicity local galaxies in emission line ratio diagrams involving only lines of hydrogen, oxygen, and sulfur, but show a systematic offset in diagrams involving [N II]\(\lambda\)6584. We conclude that the offset of \(z\sim2.3\) galaxies from the local star-forming sequence in the [N II] BPT diagram is primarily driven by elevated N/O at fixed O/H compared to local galaxies. These results suggest that the local gas-phase and stellar metallicity sets the ionization state of star-forming regions at \(z\sim0\) and \(z\sim2\).