It is basically true that the modern word for "crisis", wei2-ji1 or wei1-ji1, is analyzed as "danger" + "opportunity".

Maybe the most appropriate thing at this juncture is to discuss the meanings of the second morpheme, "opportunity". I will concentrate on the etymology of the character, since in my experience that is the kind of thing that most interests people, and about which the least accurate information is to be found at large.

The graph with which it is written is what is known as a "xiesheng character", meaming a phonetic compound character. On the left side is a determinative element "wood" (helps narrow down the meaning of the whole character) and on the right is a phonetic element (indicates the character's basic sound).

The phonetic element in this case is the true etymon (we might say, the "root") of the full character: it appears to be a huiyi character, combining an element meaning "fine, minute" with another element meaning "to protect by force of arms". The Shuowen Jiezi defines it as "subtle" and also "dangerous" and explains its structure as signifying "to meet with a subtle sign and resort to force of arms".

The main meaning of the xiesheng character ("wood" added to "subtle sign") )is "trigger of a crossbow", clearly related to the idea of a subtle but powerful military moment. Derived meanings include "weapon" (in general), "key object or idea", and "turning point" or "crucial moment". The twin modern meanings "opportunity" and "crisis" are both survivals of this last sense, and as you can see they conserve some of the basic meaning of the original ancient etymon.

One of the most idiomatic modern expressions incorporating this character is shi2-ji1, "time" + "trigger", meaning something like "turning point" or "key moment in time". The concept of seizing the correct moment is of great importance in traditional Chinese thought, military and otherwise. It is a little different from "carpe diem" - can anyone suggest a comparable Western classical expression?