corbeille is French for a kind of basket. And in modern a corbeille à papier is wastepaper basket. But that is not the type of basket mentioned here.
There's a lot of French in Louisiana, especially at the time of this story, which I believe is post-Civil War.
Here, the layette (baby's clothes) were probably in the cradle and the mother's valuable items were in a basket (corbeille) that gentleman is having them burned.
Ladies kept delicate valuables in these baskets. Things like gloves or lacy things.
The corbeille is probably what was called a corbeille de marriage given by the groom to the bride.