Do you use an Apple Wireless Keyboard on your Mac mini?
Have you tried to reset the NVRAM? You do this by turning on the mini and holding down the Commad Option P R keys as soon as you hit the power button on your Mac. Continue holding the keys down for about 20 seconds.
If you are unable to do this, you can reset the NVRAM by using the Terminal. Open the Terminal and type in sudo nvram -c then Enter. When prompted, type in your admin password (you will not see anything when you type this) and then hit Enter. Next, type in sudo shutdown -r now and press Enter.
After 15 seconds start your Mac and see if your USB ports work.