I have a 1963 12ft StarCraft boat, and one of my corner caps needs a replacement.
I'd like to find one, either new or used.
Does it affect the stability of the boat, or is it only for looks?
Google leads to me to the parent company (Manufacturer) of your boat, contacting them would be the first best step
Based on your photo and description you have a 12 foot boat, designed for an outboard motor. The item you are calling a "corner cap" would help hold the transom of the boat to the sides of the boat and spread the forces applied by the engine.
Additionally the handle shape in the piece makes it easier for a person to lift/carry the back of the boat. If you have a trailer, you probably do not need to worry about the handle shape as part of the design. Alternately you can mount a carry handles to the transom separately.
Answering your questions
How can I find Corner Caps for my 1963 12ft StarCraft boat?
Contact the Manufacturer and/or search used boat sources
Does it affect the stability of the boat, or is it only for looks?
It helps keep the transom (back of the boat) attached to the sides of the boat when the boat is powered by an engine. If you are going to put a motor on the boat, you need something there.