Depending on what you are trying to do with the link, it may be sufficient to drag it. If you click and drag a link in Safari, you can drop it in many places and get the desired functionality. For example, if you drop it in a text editor, it will drop the link URL (for plain text) or a formatted link using the title from the page. If you drop it on your desktop, you will get a webloc file. If you drop it on the Safari URL bar or tab, that tab will load the link.
If you really need to copy the link, one possibility is to use spotlight as a easy-access text field. Start dragging the link, hit Command+space (or whatever you have it set to) to pop up spotlight, drop it in the search field, and copy.
I can confirm that I see the identical problem with Safari 7.0.2 (9537.74.9), with all current Mac OS X Mavericks updates installed. (Thousands of request packets per second with the same kind of content as described above.)
However, while this may or may not help the original poster, I have found that this problem only occurs if the Windows server has Integrated Windows Authentication (also known as NTLM Authentication) and Negotiate Authentication enabled.
The server then sends these two headers:
WWW-Authenticate: Negotiate
WWW-Authenticate: NTLM
Safari will reply:
Authorization: Negotiate YEgGBisGAQUFAqA+MDygDjAMBgorBgEEAYI3AgIKoioEKE5UTE1TU1AAAQAAAAUCiGIAAAAAGAAAAAAAAAAYAAAABgGwHQ8AAAA=
And from there, the loop will get going.
But if Negotiate Authentication is not enabled on the server, there will be only one WWW-Authenticate header:
WWW-Authenticate: NTLM
And Safari's reply will be something like:
Authorization: NTLM TlRMTVNTUAABAAAAB4IIAAAAAAAAAAAAAAAAAAAAAAA=
This will work just fine. Essentially, it seems that Negotiate is broken in Safari, and since the server sends Negotiate first, indicating a preference for it, Safari will try it and enter an infinite loop that prevents it from falling back to NTLM.
So, if the server administrator can be persuaded to turn off Negotiate in the authentication settings, the problem may be solved.
I might add that Firefox sends the "Authorization: NTLM ..." header regardless of whether the server provides Negotiate in addition to NTLM or not. Presumably, Negotiate is not implemented in Firefox.
Update
Safari 7.0.3 (9537.75.14) still exhibits the same problem.
We previously reported the issue as a bug at bugreport.apple.com, but the bug was closed as a duplicate of a previous bug—the contents of which we cannot see, except that it is still marked as open.
Update 2
I can confirm hauns's finding that the authentication works with Safari 7.0.4 (9537.76.4).
Update 3
This issue is back in Safari 7.0.5 (9537.77.4)
Update 4
This issue is still present in Safari 7.0.6 (9537.78.2), as noted by hauns, with cifs or smb volumes mounted.
Best Answer
Unfortunately, you can't without editing some plist files. Please refer to this older question (not marked as dupe since the overall issue/question was different) for a few answers on this issue.