We present near-infrared spectra of seven radio-loud quasars with a median redshift of 2.1, five of which were previously known to have Ly alpha nebulae. Extended [O III] lambda 5007 and H alpha emission are evident around six objects, at the level of a few times 10(-16) erg cm(-2) arcsec(-2) s(-1) within similar or equal to 2 arcsec of the nucleus (=16 kpc in the adopted cosmology). Nuclear [O II] lambda 3727 is detected in three of the five quasars studied at this wavelength and clearly extended in one of them. The extended [O III] tends to be brighter on the side of the nucleus with the stronger, jet-like radio emission, indicating at least that the extranuclear gas is distributed anisotropically. It is also typically redshifted by several hundred km s(-1) from the nuclear [O III], perhaps because of the latter being blueshifted from the host galaxy's systemic velocity. Alternatively, the velocity shifts could be due to infall (which is suggested by linewidths similar to 1000 km s(-1) FWHM) in combination with a suitable dust geometry. Ly alpha/H alpha ratios well below the case B value suggest that some dust is present. Photoionization modelling of the [O III]/[O II] ratios in the extended gas suggests that its pressure is around or less than a few times 10(7) cm(-3) K; any confining intracluster medium is thus likely to host a strong cooling flow. A comparison with lower redshift work suggests that there has been little evolution in the nuclear emission-line properties of radio-loud quasars between redshifts 1 and 2.