We present near-infrared 1.5-2.5μm moderate-resolution long-slit spectra of two Seyfert 1 galaxies (NGC 1097 and 1566), three Seyfert 2s (NGC 1386, 4945 and 5128) and one Seyfert 1.5 (NGC 3227), both parallel to an ionization cone or jet and perpendicular to it. The spectra are used to study the spatial extent of the line emission, integrated masses of excited H2 and the excitation mechanisms of interstellar gas. In all three Seyfert 2 galaxies, [FeII] is found to be stronger than Brγ or H2 1-0 S(1), while in the Seyfert 1 NGC 1566 and the Seyfert 1.5 NGC 3227 Brγ is the strongest line. Broad Brγ originating from the broad-line region (BLR) is detected in three galaxies (NGC 1386, 1566 and 3227), of which NGC 1386 is optically classified as Seyfert 2. In these galaxies [FeII] is narrow and may be X-ray excited. In all galaxies except in NGC 5128, the spatial full width at half-maximum (FWHM) size of the H2 1-0 S(1) nuclear core is larger perpendicular to the cone or jet than parallel to it, in agreement with active galactic nucleus (AGN) unified models. The values of nuclear NH2 are higher in Seyfert 2s than in Seyfert 1s, with the Seyfert 1.5 NGC 3227 lying between them. The exception to this is the Seyfert 2 NGC 1386, where two extended regions are detected parallel to the cone. Coronal lines are detected in two galaxies, NGC 1386 and 3227.