STO's website required Java and Flash. Check your system to make sure both of those programs are up to date. I routinely get on using Chrome, Firefox, and my Tablet without the login issues you're discussing.
STO is about my Liberated Borg Federation Captain with his Breen 1st Officer, Jem'Hadar Tactical Officer, Liberated Borg Engineering Officer, Android Ops Officer, Photonic Science Officer, Gorn Science Officer, and Reman Medical Officer jumping into their Jem'Hadar Carrier and flying off to do missions for the new Romulan Empire. But for some players allowing a T5 Connie to be used breaks the canon in the game.
Same thing is happening for me when I try to access the "My Account" page via the link at the top of the website. Firefox takes me to a blank page and using IE to check from a separate browser, the page returned an HTTP 500 error.
Man is a gaming animal. He must always be trying to get the better in something or other.
Having read this I checked on the "My Account" link at the top of the page and I got a blank page. I'm using Firefox 23.0.1 with Java and Flash up-to-date. I checked the main PWE site which worked fine and I was able to check my PWE account from that site. I then tried the "My Account" link at the top of the STO page again and this time it loaded up fine. Maybe they've been doing some work on the site or something? *shrug*