SNMP appears to be on by default. I bought a new Airport Express recently (the white Apple TV looking version), and just ran this command against it:
snmpwalk -v 2c -c public -O e 10.0.1.1 and it returned the huge list of OIDs/values I would have expected.
You are correct that the community cannot be changed with the 6.* version of the AirPort Utility app, but 5.6.1 is still compatible even with Mountain Lion. One Mr. Frank Tisellano created an Automator Workflow to install 5.6.1. After installing this, you'll have both versions of the Airport Utility side-by-side. Using 5.6.1 will allow you to customize the community string, and the other advanced options that were lost in version 6.
See this answer to install Airport Utility 5.6 on Mountain Lion.