As long as water temperatures remain above 50 degrees, it's alright to keep your pond running throughout the winter for cold-hardy fish like koi. Tropical fish will not survive winter conditions in most parts of the US, so they should be moved indoors as temperatures start dipping lower, as should tropical aquatic plants. Even cold-hardy plants may naturally move into a dormant stage if the days get shorter in your area, so it's a good idea to trim them back. In water over 50 degrees, your cold water fish will remain active, so keep feeding them, keep your pump running, and monitor your pond's chemistry as you do the rest of the year.
If your mild region experiences an unexpectedly severe cold snap, you'll want to take steps to protect any exposed equipment. Your pond shouldn't freeze over, since water temperatures are much less volatile than air, and your fish can happily move to warmer water layers at the bottom of the pond. However, if the air temperatures drop precipitously, water in exposed above ground pumps, like the one for your waterfall, may freeze. This would be bad news since ice can damage and destroy the delicate valves in your pump. If you're expecting an arctic blast, it's time to quickly disassemble and drain pumps and stick them in the garage for a day or two.
