Site running with mysql replication, shall we shut down the site when replication broken?
Replication may fail because of different factors. You don't want to shot down the site each time replication fails. Instead, you should think of a solution for this situation: when replication fails, how to keep the site up.

It seems that you have master-master setup for the replication. I suggest to let users (application) connect to the closer server when they are reading, and perform all writing operations on one master, call it "active master". This way, you gain the speed of reading, and you know that the replication is going in one direction. If the replication fails, you fix it while the site is still up.

On the other hand, if the master fails, you switch masters: pending master becomes the active master, and you fix the failed one, then resume the replication.

Having said that, there are some situations where you may want to shut down the site. 

HTH