Hello there, Charles.
Calendar alerts in Mavericks should provide a notification even if the application is not running. This Knowledge Base article offers up some good review information on setting up an alert:
Additionally I would review Notification Preferences to see if something there is set to not alert you:
If the issue persists, you may want to test the Calendar app by using another user account:
Thanks for reaching out to Apple Support Communities.
I depended on them too. Can't believe that nobody knows how to fix this. I grieve for the loss of my G4 titanium. I wanted a computer that worked without the dreaded blue screen. I didn't want the cloud. I didn't want to sync with other devices. I didn't want targeted advertzing from a relationship with google. And i think i'm careful enough to use java and pdf's without the machine freaking out. I wanted a computer that worked doing basic things, like calendar notifications. ***!