I completed the second of the two stair baskets I was asked to make for a friend. They turned out well. I chose to use 1" flat cane as the material. While we were on holiday a few years ago in Vermont, we visited a basket-maker who specialised in making stair baskets, using a 1" birch veneer. These are modelled on that remembered example.
My memory suggests the birch veneer baskets were molded around a frame rather than made free form as are mine. I believe the rim was stapled rather than lashed. I did secure the handle to the basket with the use of brass screws as shown here.
The baskets sit nicely on our stairs. We use two stair baskets ourselves. In the winter-time we drop out gloves, mitts, scarves and touques into them, and our summer hats there in the summers.