Once you have woven as much of the basket's body as it seems safe to without it collapsing, you may need to allow the basket's lower portion to dry and stiffen. You want the clay to be stiff enough to hold its shape when it is turned over, but not yet fully leather hard.
Loosely wrap the upper areas that are still under construction with plastic. Plastic grocery bags can work well. Depending on humidity, heat, and how wet the clay was when you began, drying could take an hour or more.