There is other observational evidence that supports the premise that quasars are at the distances implied by their red shift -
1. Lyman alpha forest: inter-galactic non luminous gas clouds cause absorption lines in the spectrum of quasars. The number of absorption lines increase with distance, are less redshifted than emission lines of quasars, and do not appear in the spectrum of objects less distant than these gas clouds.
2. Gunn-Peterson trough: shortly after recombination [~300,000 years following the big bang], it was predicted remnants of the primordial soup would still floating around for a few billion or so years. These remnants would selectively block certain wave lengths in objects sufficiently distant. In the last year or so, the GP trough has been found in some extremely red shift distant [z=6+] quasars.
3. Type Ia supernova: Certain, extremely bright supernovas are useful as distance indicators because they have similar luminosity and their brightness decreases at a consistent and predictable rate. Type Ia supernova observed in galaxies at distances comparable to quasars have the luminosity and periodicity expected based on their own redshift, as well as the galaxies in which they occur.
There are additional indicators that remote objects, such as quasars, are at the distances suggested by their redshift, but, these are among the most compelling.