Here's a picture of my mint plant showing a flower (left) and most of the plant (right). The way I harvest is to pick a few leaves (how much ever I need), but never all from the same stem. I pick the biggest leaves from different parts of the plant, and by the time I need more, the plant will have produced lots more. As you can see, I have way more than I need.
However, if you find that your mint requirements are generally more than what you can reasonably take from the plant at once, I would suggest getting a second or a third plant and cycling between them. That way, you pick from the first, then from the second the next time, then the third and so on. By the time you come back to the first, it would have produced more leaves. I also explain this in a related answer pertaining to cilantros.
Click to enlarge
Now coming to the leaf "drying" up, it could be because it has bolted or because the plant is rootbound. However, I do not think this is the case in your plant above. Now, there's no way for me to tell if it's rootbound or not, but the stem definitely does not look dried up. In fact, it looks healthy to me.
I believe the thin "dried" stem that you see in the right is a stolon, which is how mint propagates. Stolons generally spread horizontally along the ground surface. However, by their very nature (technically being stems), they seek out light and since there is no way to grow horizontally in a pot, they grow upwards. In my plant, since it's in the ground, the stolons just spread outwards, instead of upwards.
Another identifying feature of a stolon is that they have longer internodal distances. What I mean by this is that if you look at the distance between the "notches" (where the leaves form) in the thin stem on the right and in the other thicker ones, you'll see that it's longer, which further confirms my suspicion.