As you state that you are using OpenMPT 1.19.04, it should actually not report any new versions since that is the most recent one, so there should not be any dialog at all.

I have just tried to reproduce this with an outdated version and as expected, it just shows me the dialog once.

Have you probably installed OpenMPT in "portable mode" to your program files directory? In that case, OpenMPT might not have the necessary rights to write its own configuration file, so it will "forget" that it already checked for updates every time. It will not be able to save any other settings in that case as well. If that's the case, you should reinstall OpenMPT in normal mode.

I did not install OpenMPT in "portable mode". The OpenMPT configuration file is located under C:\Users\Keith\AppData\Roaming\OpenMPT and it gets updated there as expected.

However, I noticed that today (the first day after installing OpenMPT 1.19.04) the update dialog no longer appears. So it seems that the issue here is that OpenMPT keeps prompting repeatedly during a day when it is supposed to prompt, rather than prompting just once.

As you also have an issue about random crashes in OpenMPT open, is it possible that every instance of OpenMPT which showed the update notification crashed? OpenMPT only writes application settings when closing the program, so if it crashes, it doesn't get a chance to save the flag which is used for the update notification.

No, I cleanly exited OpenMPT several times during the day yesterday (no crash) and yet I still always got the "check for updates" dialog every time I started OpenMPT again.

That's really weird. Just to be sure, do you mean the "OpenMPT would like to check for updates now, proceed? [...]" confirmation?
It's supposed to be a one-time thingy and from the code I cannot really see how it would be shown several times...

Correct, that's the dialog I'm referring to.