Thanks to Jeffrey Lawrence, significant changes have been made to the DNA Master code which now allow it to access the newly-secure servers of NCBI. To obtain these changes, you need to update the program. Without updating, important features of DNA Master will not work, including auto-annotation and BLASTing.
To update, open DNA Master then go to Help and click on Update DNA Master. (You will need an internet connection for this to be successful.) You will see a message along the lines of "Update successful upon restarting DNA Master."
After restarting, the following window will appear.
This confirms that DNA Master has been updated to include secure connections. There is also a new Secure Connections panel in the DNA Master Preferences. See below for the suggested values. (These should be the defaults.)
To test that your DNA Master is properly connecting to the new secure NCBI servers, you can try auto-annotating an imported fasta file, then BLASTing one feature from that auto-annotation. If both work, your DNA Master is ready to roll for another year of annotation!
The updated DNA Master is version 5.23.0, build 2487, 15 Nov 2016.