1. In Korean culture, the half-moon is a symbol of growth.
2. In Korean culture, the half moon is a symbol of growth.
(Are both OK? What about the use of a hyphen between 'half' and 'moon'? And what is the use of 'the' before 'half moon'?))
WITH the hyphen: http://www.answers.com/topic/half-moon
The word "the" is an article used to modify the noun "half-moon." It indicates that you are referring to a particular half-moon.
http://grammar.ccc.commnet.edu/grammar/determiners/determiners.htm
Both sentences are correct and convey the same meaning. The use of a hyphen between 'half' and 'moon' is not necessary in this context as 'half-moon' is not being used as a compound adjective modifying a noun.
The inclusion of 'the' before 'half moon' indicates that the particular half-moon being referred to is a specific one, possibly a commonly recognized symbol in Korean culture. This usage suggests that there is a specific half-moon symbol that is universally associated with growth in Korean culture.