I've had this problem as well, and disabling Google labs features seems to help. There are a number of ways to do this, but the easiest for me was to replace "#inbox" with "?labs=0" in the URL in the address bar (this was easiest, because I have multiple accounts, and if you do too and I give you a complete URL, it could be for the wrong account (the number following "u/" in the address bar) and you'd have to edit the URL anyway).
With labs disabled, I was able to download. Then I changed my gmail settings to disable all lab features except the one or two most important to me - downloads still work with labs otherwise enabled.
In short, I wasn't able to pin down which Google labs feature was the culprit, but it seems like one was....
Good luck!