Your iPhone has "No backlight problem". Usually caused by water exposure.
Try a Reset, Hold both Home and Power buttons until the iPhone restarts itself.
If still problem, you could try a Restore of Firmware with iTunes.
First step in Fix (less common problem), is to replace the LCD Display. With iPhone 3GS before opening and during repair make sure the iPhone is OFF or you risk further damage.
If still problem, then the cause is (most common problem), faulty Logic Board Display Chips, most likely the Coil. This Coil can be replaced by someone very experienced with soldering.