Binaural Beats meets Sound-Color-Synesthesia
Unfortunately, it appears there is currently no research investigating how synaesthetes experience the binaural beats effect. If any such research does exist, it does not appear to be available (in English) via Google Scholar, Web of Science or Scopus. 

Furthermore, there is little research on binaural beats generally. I covered some of the scarce research that does exist on binaural beats in my answer to the question, "Is there scientific evidence on the benefits of binaural beats?", which may be of some interest to you. However, the findings are generally inconclusive and do not allow us to theorize on how a synaesthete might experience the binaural beats effect. On the evidence, therefore, it is currently equally conceivable that synaesthetes do or do not experience binaural beats differently from healthy participants at all.