Is this sentence punctuated correctly?
Not only am I a wife, but a mother; not only a daughter, but a sister; not only a coworker, but a friend.
Consider a slight re-wording (and punctuation) of the sentence as follows:
I don't quite understand the logic of the sentence. That is probably irrelevant, as it isn't what you asked about!
No. However, the problem doesn't originate with the semicolons but with the commas. I see what you're trying to do, though. You have a series of items with their own commas, so you're using semicolons to separate them. This would be correct except the items here don't need commas.
The correct punctuation is as follows, along with the missing words. I also had to revise the first few words to solve a parallelism problem.
Why no commas? "Not only...but also" is a correlative pair. Correlative pairs aren't divided with commas. See "Commas with But Also": http://zencomma.wordpress.com/2011/06/18/comma-with-but-also/ for more information about correlative pairs.