Kanji: 駄.

「駄目」の駄と「下駄」の駄は同じ漢字だみたい。面白い。どうしてんだろう？　

The radicals are a horse (馬へん) and fat/thick (太い)。先生 told me that often kanji that share a tsukuri will also share the 音読み of that tsukuri—so I think TA must come from 太.

ヘンシャー先生の本に調べよう*。。。。 Couldn't easily find in Henshall because it's filed as DA, a reading that I thought was primarily rendaku; Henshall doesn't even include TA. But Kotoba is very well cross-referenced and leads me to Henshall 1541.

Kotoba gives a primary meaning of burdensome, and the compounds it lists tend to be about foolishness/worthlessness or shoddiness. Henshall says it means a pack-horse, written with 太　or 大, and that because pack-horses were cheap it took on that meaning, too.

So do you suppose geta is written 下駄 because geta are under you and support your weight, or because they're under you and they're cheap? 「駄目」の表現の場合には、よく使う言葉だから、漢字の意味か音が重要だかも知れない。 I guess the sense of foolishness works, but 目 may be there just phonetically.

Here's something fun I discovered while looking up 駄目: damemoto (駄目元)—which Kotoba defines as giving something a try because one has nothing to lose. 「する」と使われるかなあ。**

*Originally I was going to use 相談する instead of 調べる, but I couldn't decide on "を相談する" or "二相談する".
**That's me trying to use the passive form of 使う. 駄目元したね。

Google has let me down; 下駄を張っている動物の写真がないんだ。