Question form an inexperienced scanner.
How does rain affect the scanner antenna ? How does it affect reception ?
Unless the antenna is damaged and water is collecting in coils or around insulators (usually not a challenge for RX only), usually rain, on its own, does not impact the antenna function.
Depending on the frequency, weather patterns can impact propagation.
What does everyone do to "waterproof" the antenna itself, and any cable fittings ?
A properly designed antenna should not require any additional waterproofing. If it's an antenna you built yourself, you'll need to look at it and decide what parts need to stay dry.
As for waterproofing connections, that should always be done for all outdoor connections. Water + Copper make a real mess and will destroy coaxial cable and connectors.
There's a lot of ideas about how to waterproof coax connections, and some of those are good, some of them are terrible. At work, we use the industry standard methods that have been adopted by damn near everyone who does this professionally: