We use high spectral resolution (R > 8000) data covering 3800-13000r{A} to study the physical conditions of the broad line region (BLR) of nine nearby Seyfert 1 galaxies. Up to six broad HI lines are present in each spectrum. A comparison - for the first time using simultaneous optical to near-infrared observations - to photoionisation calculations with our devised simple scheme yields the extinction to the BLR at the same time as determining the density and photon flux, and hence distance from the nucleus, of the emitting gas. This points to a typical density for the HI emitting gas of 10$^{11}$cm$^{-3}$ and shows that a significant amount of this gas lies at regions near the dust sublimation radius, consistent with theoretical predictions. We also confirm that in many objects the line ratios are far from case B, the best-fit intrinsic broad-line H$alpha$/H$beta$ ratios being in the range 2.5-6.6 as derived with our photoionization modeling scheme. The extinction to the BLR, based on independent estimates from HI and HeII lines, is A$_V$ $le$ 3 for Seyfert 1-1.5s, while Seyfert 1.8-1.9s have A$_V$ in the range 4-8. A comparison of the extinction towards the BLR and narrow line region (NLR) indicates that the structure obscuring the BLR exists on scales smaller than the NLR. This could be the dusty torus, but dusty nuclear spirals or filaments could also be responsible. The ratios between the X-ray absorbing column N$_H$ and the extinction to the BLR are consistent with the Galactic gas-to-dust ratio if N$_H$ variations are considered.
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