Blowing on the cart sometimes worked, if there was errant dust or fluff in the way. But in the long run the salty moist air would cause a corrosive layer of gunk on the contacts. Normal procedure was to open each cart and clean them with a good eraser then an alcohol wash.
The bigger problem was bad design in the original NES. The cart plugged into a component that looked like a harmonica and then pressed downwards, levering against a pretty heavy spring. This perpendicular stress caused the contacts to break, first causing contact flakiness, then failure. The SuperNES and GameBoy fixed this problem with a vertical insertion that carried over into the N64 and others.