I experienced the same problem where by I could not log into my Nationwide Bank account when I had upgraded Safari to the latest Version before Mavericks 6.1.0 on OSX Mountain Lion 10.8.5. I believe.
I already had Trusteer Rapport installed as recommended by banks.
Some Banks worked but Nationwide skipped the log in page and kept taking me to a page about Cookies telling me I needed to enable them but they were enabled. I just could not get into the Log In Page. Incidentally the Nationwides Cookies screen shot is of an out of date Cookies setting page - it is now under privacy in Safari preferences.
I then installed Rapport again when it popped up as an option and bingo I could get to the Log In page.
Took the plunge and upgraded last evening to Mavericks
Same probelm occured cannot Log In to Nationwide.
I have just stopped Trusteer Rapport and thank goodness I can get into Log On again.
However when I logged out and tried to open another site Safari crashed with the following error message
"safari quite unexpectedly while using trf.dylib plug-in"
Chose Reopen and safari opened Ok
I googled the message and that got me here - seems it is Trusteer Rapport
So someone needs to sort this out I guess Trusteer I guess?
Hope this is helpful
Cheers
John