If you want to say that something has been written down, you must say "written" because it was done in the past. If you want it to be written down, You can say "I want this to be written down." Or "Please write this down for me,"
Here is the strange thing about 'to be written'... You are saying, though it is not written down already, in the future you want it to have already happened or finished (the act of being written.) If you want to see more about this, it is called a future perfect tense.