There is a great diversity among contemporary Catholics in the posture one assumes while praying the family Rosary. Many take the traditional posture of kneeling, a posture of humble supplication. This is a commendable practice for many reasons. I often wish we did it.
On the other hand, many of us either sit or stand for the Rosary. One family we know stands because they are steeped in Byzantine spirituality. Byzantines stand to pray. We ourselves tend to sit. In our culture, the typical posture for meditation is sitting, so it is appropriate to sit when one is meditating on the mysteries.
I tried for a while at one point to have us kneel, but it didn't work that well. It never "took." It was too hard to concentrate on the mysteries. On on the other hand, I wanted us to do something to show physical reverence beyond sitting our our duffs. The theology of the body has made me even more sensitive than I was before to the need to engage the body in our spiritual life.
So, I reflected upon the fact that the Rosary was historically a way that illiterate lay people could participate in the liturgy of the hours. My experience at monasteries is that the monks sit during the psalms, and stand for the Gloria Patri (bowing) and the Magnificat. Our current practice, then, is to stand and face the image the Mother Thrice Admirable from the Creed to the end of the first Glory Be, sit for the Our Father and Hail Marys of each mystery, stand for the Glory Be, and stand from the Salve Regina (which we sing in Latin) to the end. Our current practice adds a sense of reverence, while allowing us to sit for meditation, a proper posture for that purpose.
Another rosary practice we have is to occasionally allow our younger children to draw the mysteries while the rest of us are praying the prayers. Over the years they've become quite sophisticated in depicting the subtle significance of each mystery. This has helped us all appreciate the depth of the prayer better.