ASDM itself should be working with Java 7, though we don’t officially support it and no formal testing has been done to verify that it works properly. The issue that customers are running into is with ASDM Launcher, which was recently updated to work with Java 7. Development team is considering in posting a new maintenance release for 6.2F that would have the new launcher bundled into it.
In the mean time, as a workaround, users can get the latest version of the launcher by installing ASDM demo for the latest ASDM version: http://www.cisco.com/cisco/software/release.html?mdfid=279916878&flowid=4819&softwareid=280775064&re.... The install will update the launcher files. It will not change the ASDM image the customer is using, because the image itself is located on the FWSM, while the demo files are installed locally on the user’s computer.