If you haven't done it already, I would highly recommend that you watch and get into discussions of the materials the Church has for preventing the sexual abuse of minors. You will learn a great deal about what to look for in order to keep your child safe with all adults, not just the few you know to be a danger, and without making your children or yourself into paranoids. Don't just watch the stuff intended for parents. Take the training intended for adult volunteers, too. I'm a volunteer catechist, and I've learned a lot.
Free training on how to recognize and deny predators the opportunities they need to perpertrate their crimes and how to give help our children avoid being victims without ruining their innocence is the best thing to have come out of this whole horrific mess. Don't miss it. Take advantage of it.
It would not hurt if all the adults in your family learned the rules that catechists all have to follow now. They are intended to a) deny predators access to children, both by educating adults about what opportunites are needs and educating kids about their right to say "no" and to report behavior they're not comfortable with b) allows adults to show affection without allowing grooming behaviors, and c) tells adults how to avoid circumstances where they will be prone to credible accusations. In other words, it protects adults, too, since children who have suffered real abuse have been known to level accusations against an adult other than their true abuser. The programs also helps children to identify * many* adults they know they are allowed to talk to about incidents that even make them uncomfortable, including someone outside the family and even when the person they have a problem with is in a position of trust and esteem.
Yes, I am saying that one of your young relatives who has been abused by some other family member might accuse you, instead, if you even allow yourself the opportunities that abusers need when you are with other people's children. More to the point though: when trusted adults act one way and perpetrators have to act very differently in order to create an opportunity for themselves, the difference in behavior is more likely to raise a red flag in a child's mind. This is a good thing.
As for the family member who is a convicted predator, the family needs to let them know that if they don't stay within very strict boundaries at family events, they will be asked to leave. If the other members of the family don't go for that, then don't go to the events. By strict boundaries, I mean the kinds of boundaries you would expect at events at which your child would be surrounded by total strangers. In that family, frankly, I would expect every adult other than a child's own parents to observe those boundaries. You don't know who has been abused, or who has been made into an abuser.
IOW, this individual may not be an isolated case in the family, as sexual abusers often suffered abuse themselves in childhood, abuse which is usually not disclosed. Sexual abusers can also still be in childhood themselves. Please learn about this danger to your children; you will never regret that you did. Encourage all of your relatives to do the same. This isn't to demonize the person who committed the crimes. It is so that there won't be another crime or even grounds for another credible accusation in your family. If this predator was the product of a cycle of abuse, it is time for the cycle to be broken. If not, then the behaviors are still healthy ones for children to learn to expect of adults caring for them. It is a no-lose proposition.