applicationsfilesystemmacos
I often get the dialog box warning me that I'm opening an application for the first time. But, it isn't the first time. Some of these are applications that I use daily. 
I seem to be getting more of these after an OS upgrade, almost as if some database got reset. Can I do something to get these warnings to be generated properly?
It sounds like there is a permissions error somewhere. Run Disk Utility (from the /Applications/Utilities folder), select your boot drive and click on the "Repair Disk Permissions" button. I usually keep pressing the buttons until there are no errors left. Reboot and try running one of your problem apps. It may warn you again, but after that it should remember that you have run it before.
Good Luck.
You can disable the startup items entirely, so they don't attempt to run at login.
System Preferences → Users & Groups → (Kid's account) → Login Items
Select the applications that are denied, then click the minus sign to remove them from the boot process.
See also: http://www.maclife.com/article/howtos/how_remove_startup_items_os_x
Best Answer
This warning pops up every time an application wants to register a URI handler (a
something://protocol). Normally, your approval is saved by LaunchServices, but that database can become corrupted. Try resetting it by doing ain Terminal. The warnings should pop up once for each application again after that, then be gone for good. If they keep coming, something is resetting the database on its own (a clue would be that the default browser / mail app / RSS app / FTP app you set keep resetting to the Apple provided ones).