Fear has its place, of course. Without it, we'd be much less likely to avoid dangerous situations. Without fear, we might be more inclined to step suddenly in front of oncoming traffic; to dive from great heights into murky, unfamiliar waters; to walk blindly where poisonous snakes and spiders are plentiful. Indeed, according to this, a Buddhist perspective on fear and phobia, fear has been central to the rise of civilization:
...we fear discomfort so we store food for more difficult times, we prepare ourselves for dangers like wild animals, or to defend ourselves from other humans. This fear of discomfort and attachement [sic] to comfort has driven humans in their development from a type of smart monkey to a creature that has gained control over nearly all other living beings on this planet.
In some cases, though, fear becomes irrational. This is the world of phobics, the chronically anxious, and people who suffer panic attacks.
As it turns out, Western psychology's strategy for overcoming irrational fear is quite similar to Buddhism's strategy for overcoming irrational fear. In Western psychology, for instance, the therapy is to
...discover by experience that the feeling of fear (paranoia) is an exaggeration of what we perceive in the world, and force our rational mind to keep in control of the emotion. So, if you are afraid of spiders, perhaps the cure starts with simply drawing them, then looking at a small one - far away locked in a safe place - then forcing yourself to go closer (the rational mind says that nothing can happen), in the end, usually the patients will regain so much control that can even hold a poisonous, hairy, huge tarantula in their hands - obviously the end of therapy! This is not because they are exceptionally brave people, but they have gradually learned to take control over their exaggerated emotions, by realizing these emotions were not based on a real danger.
The Buddhist therapy of treating exaggerated fears is...not essentially different from the Western ways of treatment. Treatment is based on trying to see that fear is a form of suffering that we wish to get rid of, and using habituation and the control of our mind to dissolve irrational fears.
When it comes to overcoming fear, of course, there are some major differences between Buddhism and Western psychology. The big one: Buddhism claims to teach a path to extinguishing fear of all kinds, not just irrational fear, by going beyond just overcoming specific fears to the root of all fear: "...Buddhism tries to take the solution of mental problems to their very end, to stop our very potential for suffering..." Think about it: you can do CBT as much as you want, but you're never going to get rid of your fear of death. Buddhists, on the other hand, claim that by following the path of Buddhism you might do just that.
According to the Buddha, all suffering (including fears rational and irrational) is the result of attachment -- attachment to our "self," to the people in our lives, to our possessions, to our way of life, to the things we know. Because all these things are transient, to the Buddhist, being attached to them makes no sense. Through Buddhism, the claim goes, we can learn to give up our attachments. The freedom that results is nirvana. We still get sick, we still get hungry, we still get tired, and we still die -- but these things no longer cause us suffering.
From where I sit, whether that's true or not remains to be seen. But there are a couple of benefits of Buddhist meditation that are clear, from my experience. One is that there's calming value in the perspective that suffering is an illusion, created by your mind based on impermanent circumstances. The other is that meditation -- emptying and/or focusing the mind while sitting in silence -- can work wonders when it comes to calming the troubled waters of your mind.