I have also been having this same problem ever since upgrading to Mac OS 10.14, "Mojave".
I have since found a workaround that isn't too painful. After starting up the double-clickable app, use the Mac terminal application to execute the Mac Console utility:
open ~/Library/Logs/sage.log
At the end of the log file, you should see a message:
Copy/paste this URL into your browser
when you connect for the first time,
to login with a token:
After pasting the URL into your browser, you should immediately see the notebook interface, and you can then use Sage. Despite what the message says, you will still have to do the copy/paste operation on subsequent Sage invocations.
If you wait for about a minute or two, the Sage icon will eventually appear on the menu bar. If you are patient, you can skip the copy/paste step and wait a couple of minutes for the app to open by itself.
There are also ways of opening the notebook directly from the command line, see Running the notebook.
Can you say exactly which one you downloaded and so forth? That might help us debug it.
For my part, I am using sage-8.6-OSX_10.14.2-x86_64.app.dmg on 10.14.3. But this has been a problem ever since Sage 8.4. The common denominator has been "Mojave", in versions 10.14, 10.14.2 and now 10.14.3. It looks like Sage 8.7 is about to be released, and if this hasn't cleared up, I guess I'll mention it again on the sage-devel forum.