No fix, but here are a couple of possible workarounds:

Restart your machine in Safe Boot mode (hold down the shift key right after the startup tone), run Software Update again, then restart normally. Java sometimes needs a few machine restarts to work properly after a new install.

Create a new admin user account and install LW there.

Get more info: run Applications/Utilities/Console and copy the readout when LW quits unexpectedly. Post the info here or in the developer's beta forum at LimeWire Forums