I don't really understand what one actually means when one says about doing string theory in a background independent way.
Apparently B. Sathiapalan is the only person (as far as I know from literature searching) who has been looking into it from a long time. Here is a very recent review by Sathiapalan. The introduction and conclusions of it look very nicely written.
What is background dependence in string theory and what must a background independent theory fulfill?