Interestingly I have four Due boards and three different computer systems and have not had any USB connection problems. They have worked reliably the first time I plug in. There are exceptions I have been experiencing with the Due Native port not working with some USB cables. But the Programming port has never had any problems.
It is sound like you may have some kind of hardware problem. You need to narrow down the possibilities by replacing the USB cable, Computer, or Due board. Which USB port on the Due have you been testing with?