Also make sure that the lid to the PSX is completely shut. Sometimes it may look shut and feel shut but you may need to apply a bit of pressure to the lid. Also ER is right, if no game is inserted then the PSX goes to the memory card screen.
Be careful not to use alcohol on the eye as it could damage the circuit board. If even a tiny drop gets on the board it could mess it up pretty bad. I'd say to use a rag, or possibly some canned air.