I check the return code from the borg backup job and issue a notification to my phone via NTFY if there was a problem.
Please note that return code 1 (warning) will be issued if a file changes while borg is backing it up so it's very common if you backup log files for example. Which is why I only notify for code 2 or greater.
You can also do it the other way and simply issue a notification no matter what happened, and just list the return code as-is. This is probably better since you also get confirmation that the backup job is actually running.