Evernote doesn’t currently support drag and drop out of its system. It’s on their list of to dos however, according to a forum post. Other options are as follows:
On a Mac, you can drag pretty much anything to Finder and it will be saved for later. PDFs appear to be an exception.
Right-click and save works well, and you can also (in Windows, anyway) copy the item and paste it where you want it to go.
You could use HTML Export to send the note to a local web page, which will drop the attachment in the folder with the HTML file.
If you use Gmail, you can email the note to yourself and then drag the attachment to your desktop (or wherever).
If you use Chrome, you can open the web version of Evernote and download the file and then drag it from Chrome to your desktop.
There us also the possibility of a browser extension that allows you to drag file links to your desktop, but I’m not sure on that one.
I hope this helps you out! :)