I'd use 认识 for characters.
If one says he 知道 a character, I'm not sure if he really knows the character (it sounds like he has only heard of that character. He knows the existence of the character but does not know what it exactly is).
I agree with fyl. In my mind, "我知道這個字" means "I know of this character." (i.e., It rings a bell. I've seen or heard it before. I am aware of its existence.), whereas 我認識這個字 means "I know this character" (i.e., I am familiar with this character. Not only am I able to recognize it, but I also know its meaning and pronunciation).