At 4AM last night my wife and I were sleeping soundly when both our phones started going crazy. She has an Apple iPhone, I have an Android, so I started digging around on our phones to find out how to turn it off. You could always turn it back on during the day if you are concerned about getting the alerts, but I am usually listening to the radio in the car and at work and have the TV on when I am home.
Also, in Kansas City, we have these giant alert signs above almost every highway that show the AMBER alert if you are in the car. Heck, you could make a case that it is unsafe to have the alerts enabled while you are in the car, because it makes a strange sound, not like a text message, so you are probably going to look at your phone to see if it is going crazy.
Anyhow, it is your decision if you want to disable the alerts, I am just showing you how if you want to disable those annoying alerts.
On your Apple iPhone it is pretty simple:
Go to Settings--Notifications and scroll down to government alerts.
There you can easily opt out of the AMBER Alert notifications.
You can also opt out of the government's emergency disaster alerts at this screen by clicking on Emergency Alerts. I personally left this enabled because if there is a tornado at 4AM I am happy to get woken up, but an AMBER Alert, not so much!