I had someone posting just bizarro anonymous stuff randomly on my posts. I leave anon posting on because I've had people who don't have accounts on the blogging service find my stuff and want to contact me. I do however track IPs. I left the posts up without comment, mainly because none of them were attacking me personally (none of my blogs have personal info in them; it's all fanfic and the like), and they were just bizarre posts. Anybody reading them would see that. Finally the person stopped doing it because she wasn't getting any attention and she probably realised that the posts made her sound foolish. 'Cause she was.
If I had a personal blog, I would probably screen anon comments and track IPs.