There are a great many ways to do this. However, this one specific style will not only be easy, but will also require materials that everyone generally has at home anyway. This particular basket has the added benefit of being a basket that will allow you to recycle both during and after the making of it.
The materials that you will need for this project are old newspapers, a foam brush, craft glue and decoupage paste.
You will need a workspace. Clear this space and then cover it with a sheet of newspaper to help protect it from damage by the glue and paste. Tear your newspaper into individual pages. Take one sheet and lay it out so that the print is sideways in front of you. Roll it up tight and then unroll it. Using the foam brush, paint it with the decoupage paste and roll it up tight again. Repeat this process with the rest of your paper.
When you get all of the paper rolled, you will take the first roll and coil it tightly to form the bottom of your basket. Where ever the coils touch each other you will need to add some of the glue in order to secure the coils. Continue with the next rolled paper by gluing one end of it tightly to the outside end of the previous one. Continue in this fashion with the rest of the papers. You will mold and shape your basket as you like making it is large or as small as you like. When you finish allow 48 hours for it to completely dry. You can then paint it or do what you like with it.