We ran into the same thing when we were using basic alerts. The only way I found to clear them was to disable and then re-enable the alert.
Have you given any thought to converting to advanced alerts instead? Advanced alerts are much more flexible than basic alerts, and if you're in an environment where you are continually adding devices and have a lot of alerts configured, advanced alerts are definitely the way to go. The ability to tie alerts to custom properties eliminates the need to touch multiple alerts to click a check-box when you add a device; rather you simply apply the appropriate custom properties to the device, and you're done. Also, I imagine the basic alerts will be phased out of the product in favor of advanced alerts at some point.