I guess you cannot really protect the fabric itself easily / beautifully.
I have in mind only one option - to be done by someone with experience:
- remove the handles (or at least one end of each);
- pull a protective tube over each handle;
- reattach the handles.
Step 2 might involve creativity: what material to chose what color etc. If you want to maintain the original appearance of the bag, then have the protective tubes (sheaths) made form some good quality black material. But red or pink will match equally well, if you are so inclined.
A very creative alternative (more like an attempted hack, rather than a solution) is to trim the frays away, and then to "paint" the handles with some kind of flexible glue - in the idea of liquid rubber. However, I do not have the experience with it, and I cannot guarantee that it will save the bag - or maybe just destroy it.
A solution which does not require any special skills is to use a wider ribbon (color of your choice) anand just wrap it around the handles. Worst case, you just make knots antat the ends to fix it there.
The down side of this solution is that it might be quite short-lived, and therefore you might need to do it repeatedly.
A slight improvement will be to sew through the handles and the ribbon, just enough to not allow the ribbon to unwrap - and thus extending the life of the hack.