Hi, Welcome to JustAnswer.com, I am glad you are here! My name is Eric & I will endeavor to answer your question.
I already said wow once tonight because I thought the customers 23 year old Stanley opener was the oldest I had seen in quite a while. Your Genie 450 is however, somewhere between 32 and 40 years old. Double WOW!
It ceased being produced long before intellicode was even a glimmer in it's inventors eye!
There simple are no longer any remotes compatible with this opener, not from the manufacturer nor from anyone in the aftermarket parts market. Unless you can find an old one on eBay, there will be no joy in searching for a replacement.
There is however, still hope! You can install a Universal Radio receiver kit. What this does is allow you to install a receiver on the wall next to the wall mounted push button and use a new modern remote which will then communicate with the receiver which is harwired to the push button on the wall. You may even purchase a wireless keypad to mount outside the garage door which will work with this system.
Genie makes several kits with various available options and they may be purchased here: UNIVERSAL RADIO RECEIVER KITS AND RECEIVER BOARDS
Blue text is clickable to go to the website.
That said however... I STRONGLY recommend that before you spend even a dime on doing anything at all to upgrade your ancient Genie 450 opener, that you consider simply replacing it. Here is why...
A. A garage door opener has an anticipated serviceable life expectancy of 12 - 15 years and by some miracle, yours has exceeded that by several times over. Being that your door opener is so old, it is very likely to soon fail you at the most inopportune time such as in a rain or snow storm thus locking you either in or out of your garage. Or even worse yet, when you need to get into your garage fast due to security issues.
B. Unless your opener has been modified at some point after it's original installation (and many of the old 450s have been), it has no safety sensors (infrared or laser beams) to make it stop on its way down if something such as an animal or small child or even an adult or a bag of groceries or your car hood etc. gets in the way of the closing door. This by itself makes it dangerous and even illegal according to current building code. It is just not safe.
C. Quality new (and safe) openers these days can be purchased for about $150.00 from the big box stores and can often be located even cheaper (& New with warranty too!) on eBay.
Last, Have I mentioned that it would be a VERY GOOD IDEA to replace your opener?