I have a Cisco Aironet access point AIR-CAP3602I-A-K9. I connected it to my laptop (running Windows 8.1 Pro 64 bit) with a USB-to-DB9 and DB9-to-RJ45 console cable. I could not open CLI via Putty or SecureCRT. After waiting several minutes, I only got a blank screen.
Did you test this by trying to access any other devices console (like router or switch). If this does not work with them then it may be this application issue rather something to do with 3600 AP itself.