The Recruit would not know unless someone tells him/her. Since your son will be leaving soon for BC, you need to discuss with him how he wants this handled since it could happen to anyone and also consider how to handle if there is a major accident or natural disaster that affects you and yours as well. You also need to be sure that everyone who writes him or who he would call knows how this has been decided because if you decide not to tell him and then Uncle Joe writes about Grandpa's funeral or the tornado that destroyed his house, that would not be good. Realize that unless his grandparent raised him, which I doubt from your My Page, he would not be able to get emergency leave for the funeral from BC or A School or later from the command where he is stationed. My son told me not to tell him until after BC if anything bad happened. Your Recruit will need to concentrate on getting through BC and if telling him would be upsetting, hold off on telling him. It depends on the length of your Sailor's A School, whether to hold off or not. At least at A School, you will be able to call and gauge how things are for your Sailor before sharing. Chaplains make all notifications of deaths at BC if you were to choose to tell your Recruit.
Things to consider: How close was your Recruit to the one who died (had an accident...)? Will your Recruit be upset with the knowledge, which could possibly cause him to be put back in his training? Will he find comfort in knowing that the loved one is no longer suffering?? Is there someone who will be writing him who will tell?
You may want to read the following Discussions to see what has been said on this topic before:
There are others, but that gives you an idea.