In my system, the current beta does not work fine without modification, as you met. But, by following the suggestion by kmyken1 (see below: this is really useful information for me), I use MS-word with Japanese input method. As he/she said, this is an unofficial and ad-hoc way. Needs the your own responsibility. For me, it really works well at present.
Essentially, I strongly believe that the Dropbox developer team must provide the way to stop automatically version upgrade survic for avoiding needless confusion as the current case.
I use ver.37.3.25. In the Activity monitor, I found several processes named Dropbox were running. If you quit/kill these processes except for the Dropbox (with icon), you will be able to use Japanese language input method on MS-WORD again!
(Some of them need to be killed again and again. Some of them, which seem to be unrelated to this problem, revive again. You will also need to restart the MS-WORD.)
This is an unofficial way. Please do this on your own responsibility.