Which of the following is correct:
1) Don't come in, I'm busy.
2) Don't come in; I'm busy.
3) Don't come in. I'm busy.
4) Don't come in I'm busy.
Of the four choices, the only one that doesn't work is 4.
is a comma splice, but acceptable informally.
is perfectly fine, but a bit musty and over-educated for many uses.
is probably the simplest, but may not express the connectedness of the two sentences sufficiently for your taste. (Note: It doesn't have to. Apposition is a form of connection.)
You might also consider
The colon indicates that "I'm busy" is a direct explanation of the command that comes before it.
But of the four I myself would go with No. 3.