I had too much time last summer so I decided to make this after seeing it online. For the base you just fold and coil strips of magazines, like you would if you were to make a magazine bowl. The sides of the basket are even thinner folded strips, which were then coiled around something, i.e. a marker etc. I kept everything together using mod podge and hot glue.
First you need to gather a good amount of magazines, some mod podge, a paint brush and scissors.
After gathering everything you need. You get to start ripping out pages of magazines. For the base I believe I cut the pages in half (creating two long pieces) and for the circles up the sides I cut them into thirds.
To make the base you do the same thing as if you were making a magazine bowl. You fold strips of magazines so that they are long and skinny. The width will be the height of your base. You first coil one strip upon itself and paint some mod podge on so it will stay.
After finishing the base I made a mixture of some mod podge and water so it would be easier to apply. I then painted the entire base until I felt it was good. This made it much more stable and permanent.
Now the little circles are what took me a long time because there are so many of them. There may be a better way to do this, but this worked for me. I folded the strips just like I did for the base and glued them so they would not unravel. Then I coiled them around a marker and glued them so once again they would keep their shape.