We're running out of memory on an AIX box running DB2 Version 9.1. We have 32GB of memory and free memory is around 1-2 GB.

Can you suggest where to start troubleshooting this problem?

It may not actually be a problem. UNIX and Linux are very aggressive in using free memory for file caching, but will give it back immediately if needed. I would calculate the memory used by DB2 for all instances and databases (db2mtrk is a good place to start) to make sure DB2 is not using more than 80-90% of total system memory.

I would also turn off file system caching for all tablespaces, except those which contain LOB objects. It would also help if you could find a really good UNIX system administrator who might be able to limit the amount of file system caching by the OS, but those people are very hard to come by.

You can also check how many db2agents you have running on the box and how much private memory they're using. You can look at the RSS column in the "ps aux" output. Add them up to see how much private memory they're consuming. Example:
ps aux | grep db2ag | grep -v grep | awk '{ sum += $6 } END { print sum * 1024 }'

If you find many db2agent processes and most of them are idle or waiting to do some work, then consider lowering num_poolagents and setting db2memmaxfree registry variable (can set it to 1-2MB). It can bring down the memory usage considerably if you have many agents that are not doing anything.

Also, take a look at the following technote:
IBM - Recommended AIX Virtual Memory Manager settings for DB2