Do you want a remedy to the chewing, or a repair for the posts?You can see Molly's handiwork of how she chewed the banisters in the background. I don't know what to do to remedy that.
To remedy the chewing I would go to a pet store and get a spray called "Bitter Apple". Spray it on the posts and the chewing should end.
As far as repairing the posts - I was about to tell you what I would do, but then I thought a little harder about that, and if you need my advice on how to repair it, you probably don't have the tools or the experience needed for what I was going to suggest (I just barely have the experience needed) and if you followed my suggestions you could just end up in worse shape.
So instead I suggest cleaning it up with a little sandpaper and then repainting it. It won't be anywhere near as noticeable if you do this.
-And then spray it with Bitter Apple to keep it from happening again.