Based on the tests done by Brian, the installation instructions were updated: First install and remove bibledit through macports, and then install bibledit from source.
I like the idea from Mark about the 'sandbox' that macports creates for bibledit. But at the same time this thought came up: Could the reason of the crash be that the tarball does not work well from within this sandbox, and expects something installed in /usr/ rather than in /opt/local, and so crashes?
A temporal workaround is to copy the entire .bibledit folder from the old to the new mac.
The .bibledit ("dot bibledit") folder is located in the home directory of the user.
On Linux this would be /home/joe/.bibledit. Copying the equivalent of this from the old to the new Mac OSX should work around the problem of the store no longer working.