There's many ways a NES can fail but the most common issue is the cartridge contacts being dirty. The easiest way to clean them is with some Q-tip and Windex. Then you rub it back and forth on the cartridge contacts (on both sides) until all the grime is gone.
If this doesn't fix the problem, find a method to open the cartridge (it usually requires a 3.8mm security bit/Gamebit), and rub a pencil eraser on all the contacts, this way you assure to remove even the deepest corrosion that the contacts may have.
If after all this process you still have a blinky screen, then the problem may be a poor connection between the console and the game cart. At this point you need to replace the 72 pin connector on the console itself, or clean and bend the pins back to their original position (which is a really tedious task to be honest).
This problem is very common on NES consoles, and the problem originates from the bizarre "Zero insertion force" mechanism they implemented on the console, which bends the pins over time.