Solder Joints
The solder joints connect the controller to the main board. If these joints are breaking, then the controller's signal is not properly read by the main-board computer. This occurs because of the regular wear the port receives when the controller is connected and disconnected over time. You can repair this by opening the console and reheating the solder joints on the pins of the port. Let them cool and reset. Add more solder if necessary. This might be difficult to do, so it's wise to consult a professional repairman.
Dust and Lint
Dust and lint get into the console and cause problems for the entire system over time. Dust can block the signal from the controller ports and even make the console read the game cartridge improperly. Using an air duster to blow dust out of the ports will clean them quickly. Blowing into the ports is not an effective way to clean them. Your lungs are not powerful enough to expel all of the dust, and your saliva might get into the port and cause more problems.
Damaged Port
Broken ports occur from accidentally kicking or dropping the console. You also can damage the port by spilling a substance or liquid on it. A damaged port is a hardware problem, and currently the Sega Genesis is not in production and small replacement parts are unavailable. You need to consult a professional to repair the port or simply replace the entire console.