If you start up QuickTime Player from the Windows Start menu (All Programs > QuickTime > QuickTime Player), you can try changing the "MIME" settings as follows:

Edit menu > Preferences > QuickTime Preferences > Browser tab > MIME Settings button

This will launch a Windows control panel where you can designate QuickTime to handler various kinds of media. You can group the MIDI entries together by clicking the Description column. (See screenshot)

I'm not sure whether Firefox will detect this change right away, or whether you have to restart it first. Does it work?

Thanks for the suggestion, but I think that the problem is with the Quicktime Plug-in, (which simply is not shown as an option among the Applications available for the "Midi Sequence" (.midi files).

I have tried to amend the preferences and MIME settings in both QuickTime and thru the Control Panel for Default Programs and Set Associations. Plenty of file types have "Use QuickTime Plug-in 7.7.4 (in Firefox)" as a selection; and, if I could find it by browsing for that "application" (not selecting the "Firefox" program nor a driver file), there might be a ready solution. I prefer to be able to open midi files in another browser tab in Firefox or Internet Explorer, (rather than activate or use one of the programs (ie. Windows Media Player, QuickTime Player, etc.). As this problem is also encountered midi files opened with Internet Explorer, I suspect that the fault is with the plug-in in Apple's recent update.

The only other solution may be to restore my computer's settings and programs to an earlier time, then not to update the Quicktime.

QT 7.7.4 use "audio/mid" MIME type only and does not understand now "audio/midi" and "audio/x-midi" types commonly used on server side. Blame Apple for that.