No, 6s are only constantly afraid when they're unhealthy. I was diagnosed with GAD a few years back. However, one thing I learned is that it's just a chemical reaction from my body and that that kind of anxious worry won't help or change any dangerous scenario, and in fact, my body will propel me to ACTION when there's REAL DANGER and all the rest is just silly beyond reasonable caution.
However, since I'm a counter-phobic 6, all of this popped up suddenly in my 20's like a disease, since I'd been facing my fears head on and taking risks in my teens and early twenties, to counter any anxiety I'd had as a child. My counterphobic nature means that until I absolutely HAD to face my anxiety and fears, I pretended (to myself even) that I actually had less than the average person, not more. I used to drive way too fast, flipped a car, did a 360 in the middle of the freeway, smacked a purse snatcher upside the head and did various other things that a lot of people wouldn't do...while I was a counterphobic adolescent. I became extremely phobic while I was diagnosed with GAD (and I had some symptoms of social anxiety).
So I have had a lifelong personality swing between anxiety and risk taking, but what I've learned are to take more reasonable risks instead of being one extreme or the other. That's a healthier six.
I've also learned to address root causes of anxiety with things like meditation and yoga.
So what 6 mainly struggles with is not real fear (I love horror movies) but anxiety or mistrust, like [MENTION=4490]Orangey[/MENTION] posted there. I've taken great strides in actually dealing with my anxiety without just avoiding it or pretending it's not there (like I did in my teens and early twenties, which is not the most mature 6 response, obviously) and one of the main things I think I deal with is my reactive mistrust to certain people or ideologies, which you've probably noticed on the forum.
Unhealthy 6s are either very anxious or very immature in their risk taking; average 6s mainly just seek security and trust through externals, like being around safe people who agree with their ideas for the most part or being deeply involved in community building projects (as one example); the highest level 6s actually develop a level of interdependent trust and peace - integrating at a healthy 9 - and find this from internal instead of from external, and still honor their basic need to live in or be supported by groups or communities, but they don't freak out if they are by themselves, because their trust is coming from within, not from the external structures like the average 6.
So my meditation and yoga and stuff is working on being a higher level 6, so my trust and calm comes more and more from within.