I wouldn't worry too much about it. To be honest, I don't think it reflects too badly on you personally, and it says nothing about who you are professionally. As a member of your team, I certainly wouldn't judge you for it. Sure, there are always going to be people that do, but that is out of your control.
Your best course of action would be to go to work and act like nothing ever happened. Don't bring it up yourself, and continue to manage your team the way you always have. If someone else brings it up, acknowledge that it happened but entertain the conversation as little as possible in a polite yet firm way. Continue to be professional, and make it clear that the personal shortcomings of your family are not what you are interested in discussing at work. You should appear as if their mention of the event did not faze you in the slightest. This will make them look and feel foolish for bringing it up and make you look like a professional person that should be respected.
With that being said, you should definitely talk to your son about it!