ASL is Apple System Logger. It is a method to change the way log files are created. Instead of a straight text file, they are key-value pairs so that you can query them like a database.
Regardless, they are just log files. They are mostly repeats of the other log files in the system.
What you need to do is figure out what is writing so much to the log files. Use Console to see if there are any that are filling will lots of repeated messages.
I don't think there would be a problem deleting them, but if you don't fix what is filling them, I think they will regrow.
Edit: Stupid me. It took me so long to write that post I forgot you couldn't get into Console. Being that many of the entries in asl logs are repeats of the others, you can probably delete the ones in asl, then use Console to look at the others and see if there are issues.
You could also go through the folders and see if there are any very large log files.
Message was edited by: Barney-15E