Control pad's are generally all or nothing devices, they either work or they're dead.
However with a console of this vintage there are a few things that can cause this kind of issue and therefor a couple things you can do to try and resolve it.
1: Clean the serial ports and plugs (the ports and plugs that connect the pads to the console) with an air duster. Air dusters are cans of compressed air with a long nozzle and are available from most computer, electronics and hardware stores. Clean both serial ports even if you're only using one.
2: if cleaning doesn't work try unplugging the power cable, removing the console's casing and touching a hot soldering iron to the base of each of the pins that connect the serial ports to the PCB (Printed Circuit Board) in tern to re-solder them without using any extra solder, if you find that you do need more solder be sparing.
You can see a pictorial guide to disassembling a Sega Genesis here www.takeitapart.net/archives/sega-genesis/ click the pictures to see a larger image.
Lastly if you know anyone else who's in to retro-gaming swap a control pad with them to see if your pad works on there Sega Genesis and there pad works on yours.
If you find the pad actually is dead it's not the end of the world, there are a few around on popular auction sites, retro gaming sites and specialist game stores pretty cheaply.