Is this just for a particular contact? Or all incoming SMS?
If it's just for one specifically, then have you checked your settings for this contact? You might have switched the LED light off in the settings for this contact.
If it's for all SMS, then I'd first check the settings and see if something got switched off.
If it's not the settings, try rebooting the phone on the physical keyboard using shift+alt+delete and see if that resets it.
Okay I am going to have to differ to one of the more knowledgeable people on this forum. I use handcent for my texts and havent used the stock app since I got the DROID back in Nov. I looked and I don't remember how to get to the green indicator light in the stock app. Sorry.
As far as I know you can not shut it off with the stock app. It could be a software problem, but a reboot would likely have fixed this. Have you explored the option that perhaps the light has stopped working all together? Do you get the red light when your battery is going to die? If it does not work at all just trade it in for a new one, that would be a manufacturing defect.