Although the alert history isn't currently logged anywhere, you can set up the alerts to log their history of starting / stopping to a file, which sounds like it would do what you need.
The key to this is the "Launch an executable" event, which allows you to run any program or command. If you set up 2 of these events on each alert you want history about, then you can get history for that alert when it starts and stops. One of these events will Notify "when alert conditions start" and the other will be "when alert conditions end".
Now, the last part is how to write to a file. There are a lot of executables you can run here: batch files, other programs, etc. For a really, really easy method of doing it, just use the built-in Windows cmd.exe and Echo commands (note that if you're using Windows 9x, you'll probably need to replace cmd.exe with command.com in these examples).
The following command will log a message to the c:\alerthistory.txt file. There are tons and tons of ways you can modify this command to make it more useful.
cmd.exe /c echo $date $hour:$minute $Dest Alert Started >> c:\alerthistory.txt
This has a downside of popping up a DOS window each time a messages needs to be logged. An easy way of having that not happen is to set up your own Windows shortcut with the proper settings in it - either pointing at cmd.exe, or some other batch file or logging application of your choice.
If you have any issues with getting this working, please reply back here or send a message to firstname.lastname@example.org
with a description of what's not working. I just set up a test instance of PingPlotter that uses this exact setup to log starting / ending of alerts.