I don't remember off-hand, but does the Genesis boot up with the Sega screen or is it dependent on a game cartridge being inserted? If it only shows the Sega logo when a cartridge is inserted, it sounds like the problem may be inside the cartridge itself and not just a corroded contact problem. From my experience (I'm no expert), dirty contacts won't allow any part of the program to kick in. If you open the cart up, you might find some corrosion inside on the circuit board. Carefully cleaning it with alcohol and Q-tips and
letting it dry thoroughly before reassembly could easily fix the problem.
Again, I'm no expert and I'm just going off my experience with GameBoy, NES, SNES and other cartridges that ceased working. A standard red school eraser does wonders for the contacts and you can follow that up with the Q-tip & alcohol method.