My mother corrects my grammar. Still. If I send an email and incorrectly use I as an object, she will let me know (I did the other day, and she let me know about it).
Odds are, you don't have my mother. But hopefully you have someone who has instilled in you the importance of understanding how to write well. (I'm speaking primarily to writers here, but art directors who can write are awesome.)
Great art directors know their craft inside and out. They get off on serifs and kerning and leading, and it irritates them if you use the words "font" and "typeface" interchangeably. But for some reason, a lot of writers place less importance on their wordsmithing. "I suck at drawing" is a terrible reason to become a copywriter.
You have to love words.
You should get all giddy when you hear a great line of dialogue.
You should actually enjoy reading books like the one above and not feel like it's torture.
If you see a word that you don't know, you should look it up.
You should write. A lot.
You should have favorite authors, favorite books, favorite sentences.
When you read and you come across a great sentence, you should stop and consider what makes it a great sentence. How is it constructed? What is the author doing? What choices did he/she make in writing the sentence that way?
I'm not saying you have to be able to diagram a sentence (though it can't hurt). And I'm not saying that everything you write needs to be grammatically correct. But like design, there are mechanics to writing. There are reasons a sentence is strong--conscious decisions that are made in its construction. If you want to be a decent writer, you need to have, at the very least, a working knowledge of these things. Ideally, you obsess over them.
Be a word nerd. We like nerds.