It may not matter but the model # in the title is wrong. The correct one is 139.53677SRT1.

charlie try holding the remote button and then press and immediately release the learn button. I'm not sure this will work for you but I am thinking maybe it is erasing the code as you try to program it. I really suspect the remotes. Do you have a neighbor or friend with an opener you can try them on?

Don, thanks.
I tried what you said and did not work. Your comment about erasing the code makes sense because the LED stays lit for about 6 seconds (the same time specified for erasing).
I have two remotes SECURITY+ (none of them can be programmed now) and two universals that I am using for other purposes. Do you think the universals may be programmed?
If none of this works, any ideas of changing a part? I'd hate to replace the whole thing.

Well, I am assuming the remotes used to work b/c you would have had to use one of them to program your car. You say the batteries are good so I would assume that you have changed them?? When you erase the memory are you holding the button in until the LED light goes out or just six seconds and releasing? The led light must go out and then you can release and the memory will be erased. I doubt you have circuit board issue if it worked fine with your car. I would agree that you need to check out the remotes. Do you have a key pad and if so is it working fine??

Doorguy,
Thanks a lot for your input. For the erasing step I held the Learn button intil the LED went out. I don't have a keypad to check.
How can I check the remotes? The new ones are 34$ a piece in Sears and I'd prefer not to spend that money and perhaps find out that it was not the remote.

The only way to check to see if the remotes are bad is to buy a new one see if the unit accepts it or as Don suggested find a friend or neighbor who has an operator operating on the same frequency and try them out on his or hers.

Well, I bought a new remote (39.99 + tax) and it worked, so 4 years sitting in a shelf may have affected my old handhelds.
Now the universal in my new car doesn't want to talk to the new remote, the saga continues. I already programmed the car with another remote with pins that we use for a gate very easily.

Thanks to all of you for the comments and help.