Great question Leah!
I truly believe it depends on the employer. If you blog about something and mention your job, people in the job, your boss....etc...many employers would frown if they found out. In some cases, depending on what you wrote, it could be cause for dismissal if it's sensitive info.
Now,of course, all that is dependent upon them having the time to find and read your blog in the first place.
A blogger also may care if they are looking for a job. Many employers will Google applicants, if you have a questionable post, they may look negatively upon that.
It also depends on whether your job is tied in with confidential information.
A small company also might care more, for example, than a smaller organization, why, I don't know.
I would say, live your life outside of your work hours unless you have a job which specifically dictates otherwise. For example, I wouldn't recommend the president of the United States keeping a blog
I wouldn't say that keeping it anonymous is a best practice (if you're really threatening national security, they can find out who you are :)) It depends on your level of paranoia or privacy (not to offend anyone, but I couldn't think of a better word, but it's along the same premise that people are a lot bolder when they can do things anonymously). I have a totally non-job related blog and I don't think my employer could care less.
Would love to hear more thoughts!