How to make a planter out of a large rattan furniture basket?

I would just insert a large pot with a water catching saucer inside of it so that the dirt and water doesn't rot the rattan over time.

Yep. A large plastic pot that fits in the basket. It would work the best for an inside plant. If your ratten basket was outside any rain and excess moisture would ruin it. And even inside you would have to make sure that you didn't over water your plant and have water overflowing on to the ratten. Perhaps two pots, one slight bigger to catch any excess water and the smaller one would sit inside that bigger pot.

you can use the wood and create each side then screw them together. I have share a few month ago a nice example with scrap wood: https://www.youtube.com/watch?v=el7O8hC8xfk

I spray painted one of those home depot buckets off white and set it down inside of a basket I wanted to use as a planter. It has worked out great