I have an older LiftMaster opener (1/3 Hp - 1991 vintage). I have had two older replacement universal type remotes that worked fine. Lately one of the remotes stopped working with the opener. Changed the battery and the red light on the remote comes on but does not activate the door. When I push the reset button on the opener to reset all of the codes, only one of the remotes works and each subsequent time the other remote becomes operational, but not both at the same time (both remotes work but only 1 at a time not both. Any help would be greatly appreciated.

as long as your opener has a smart button any of the universal keypads made by chamberlain/liftmaster will work. such as the as the Liftmaster 387 or the Chamberlain Clicker.

You are going to have to get in there and find your problem. There isn't a common answer for this.

Here are the instructions

I suggest you look for a Chamberlain dealer in the yellow pages and explain what you have. You are looking for a 'universal receiver' that will work with what you have.

The remote and the keypad work through a signal sent to the receiver while the wall console, as you said it's wired directly to the opener, so it's likely that the receiver is shot. I'd try reprogramming the remote to just rule out the chance that all code wasn't erased first. (Press the learn button on the back of the operator head, stand back 5 feet and within 30 seconds slowly press the remote button a few times to see if it operates the opener). If this doesn't work, you can install an external LiftMaster universal receiver with the correct frequency and you should be good.
If there's a purple antenna hanging off the head, this is a 315 freq, use a 356LM, if black it's a 390 freq, use a 635LM receiver.
Then you'll have to reprogram your remote and keypad to the receiver.
Since there's also the issue with the light, there's also a possibility the circuit board got fried. Did you have any storms lately?

First thing to do is make sure it has a good battery installed. If this doesn't take care of the problem then the keypad is going bad and needs to be replaced. It is important that you get the same keypad as there are others out there that look the same but will not work.

GET A CHAMBERLAIN OR LIFTMASTER SECURITY PLUS.

DON'T GET A UNIVERSAL THEY ARE USSUALLY A WASTE OF TIME.

Both the older genie and a newer chamberlain (3-4 yrs old) both run on 390Mhz frequency. sounds like you will have to change your receivers on both operators (365LM 315Mhz) then get 3 button remote so you can program both doors on one remote.

Does your wall console have a LOCK or VACATION setting. Turn off.

It's the gear drive. There are kits online for about $40, if you're semi-handy you can replace it yourself. Search for liftmaster replacement parts.