Search Youtube for a user named SarbarMultimedia. Russ has done a pretty detailed study of the effects of airflow, and tested two different sized air pumps on his 50W/60W laser.
Episode 87:
https://www.youtube.com/watch?v=OCVsMXw6yh4
Episode 96:
https://www.youtube.com/watch?v=m4Hh-8-9Tio
Episode 116:
https://www.youtube.com/watch?v=ZKC8Byc8y28
If I interpret his results correctly, more air pressure helps for cutting, but there is a limit. However, he didn't test an actual air compressor (i.e. one with a tank), just two different sized aquarium-type pumps. I use a similar laser/pump on my 50W laser and don't have any issues cutting up to 1/4" material.
For engraving minimal/no air assist produces cleaner results.
Reading on SawmillCreek.org, some users have air compressors hooked up to their lasers, but it seems only to be required for cutting very thick materials.
-robertk925