I had this issue with a client recently during training. It was quite a twister because more than half of the laptops within the room had the option to Open with Windows Explorer available to them and there were the few random ones that couldn’t. Even more twisting was the fact that all those laptops were fresh builds using the same image. Errr!
This morning during testing some Search functionality I wanted to upload 25 documents to a document library. The option was available on the SharePoint 2013 server, was working from another laptop within my network but not from the one I was working out of.
A bit of snooping around revealed that the IE that I have got pinned to my taskbar was the 64bit version. And the ones that were being used on the server and the other laptop were running the 32bit version.
Opened the same page with the 32 bit version of IE , woop woop J!
Table below shows the current Browser support for SharePoint 2013 (Public Beta)