Actually he did take one and passed. Mia declined.
http://www.thedailybeast.com/articles/2014/01/27/the-woody-allen-allegations-not-so-fast.html
“Allen refused to take a polygraph administered by the Connecticut state police. Instead, he took one from someone hired by his legal team. The Connecticut state police refused to accept the test as evidence. The state attorney, Frank Maco, says that Mia was never asked to take a lie-detector test during the investigation.”