I left this as a comment elsejournal, about helping kids to deal with anxiety, and figured it might be useful to a wider audience. This particular analogy helps me a great deal with understanding my own anxiety and finding healthy ways to deal with it.
There are two kinds of anxiety: the solid kind and the liquid kind. The solid kind comes from one-time things you can point to and wrestle with, like a deadline for turning in homework. That kind can generally be worked around with a little bit of attention to practicalities. "I'm worried that I won't get my homework done on time." "Okay, how can we make sure you have the time and tools you need to do your homework well and efficiently?" And some explicit practice trusting yourself and recalling your past similar successes can be useful too.
But the liquid kind comes from global, environmental, persistent things: brain chemistry, ongoing physical pain or disability, a life full of uncertainty, other people around you behaving in harmful or erratic ways. It's incredibly hard to look at those things head-on because they're so big and pervasive, so we try to bottle the anxiety up in the shape of something we understand. "I'm anxious that no one likes me" or "I'm anxious that I'll fail all my classes and be a miserable failing failure" or "I'm anxious that everyone I love will die" are bottles for liquid anxiety. And you can't address that anxiety by trying to address the bottle--saying "Of course people like you, you are very successful at many things, we will all live a good long time" will bounce off the bottle and never get through to the anxiety inside. You have to pour the anxiety out of the bottle, a little bit at a time if you can't manage more, and treat it as its own thing.
In the moment, you can say, "I'm so sorry you feel scared. It's hard to feel scared. Anytime you feel scared, I will do everything I can to help you feel safe." Or maybe, "Sometimes I feel scared too. Let's hug each other and promise to help each other through scary times." You can also practice large-scale calm, meditation, compassion, patience. But the bigger, harder, most absolutely necessary work is figuring out that persistent source of fear and fixing it as much as it can be fixed.