Lately I have had someone constantly post thousands of spam blog comments which have links that point to my site. It is a huge amount of work trying to keep up with disavowing these links in Google WebmasterTools.
It's nearly impossible to keep up. Since I have to disavow manually while the spammers are automating their attack.
What can I do to better handle this situation?
According to Google, negative SEO shouldn't have an actual impact. How accurate of a statement that is is to be debated.
If they're mostly directory links or blog comments there's a reasonable chance that you're ok, even though visually it sucks by looking at your backlink portfolio. Many blog comment platforms nofollow links automatically, and other sites that don't and that are unmoderated enough that they can be systematically spammed are usually so low quality that Google doesn't pay attention to them much.