Cleaning Dirty Carts - The original NES consoles failed beacuse of two reasons. The first being dirty contacts on the game cartridges, and the second being failed or flaky 72-pin connectors. Even with a brand new 72-pin connector, a dirty game cart won't play properly. This article explains how to correctly clean dirty contacts without buying a cleaning kit from Nintendo (if you can even still find one), and without damaging the contacts.
Replacing 72-pin Connector - The dreaded 72-pin connetor is a major reason why most NES systems failed and would blink every time a game was inserted and the power was turned on. These connectors would become dirty and worn out over time, and would lead to a poor connection between the cartridge and the motherboard of the NES. You have two options regarding these connectors - purchase a new connector for about $8, or try to fix your old connector.