Since version 1.10.8 adding a mousedown handler on the Select widget is not possible, see https://jsfiddle.net/rfwmLk4t/1/ . My guess is this is a bug, because the comment near to the changed code says an expected behavior is that
3. user defined onMouseDown handler fires.
This seems to have been caused by this commit https://github.com/dojo/dijit/commit/9f5957fcba046bee274e6e8de89914c4d9f9aad6
I tested, and reverting that commit fixes the issue, but I’m not sure why it was done in the first place.
I haven’t been able to create a dojo account to report the issue properly, I would like it if someone could do so.