The usual time for Koi and Goldfish to reproduce is when the water warms in spring.
If you want to help ensure survival of baby fish, provide a safe haven for the eggs and fry (baby fish), as the adults will eat as many eggs as they can find. This safe haven can be roots of plants like Water Hyacinth, furry algae on pots or pond sides or the latest development which is soft fabric material. But think about this first, as too many fish in a pond can lead to poor water quality and oxygen issues.

The usual process starts when a male selects a particular female and starts following her around. The female will become annoyed with his too close attention and try to get away from him. Eventually, this can lead to the fish splashing about quite aggressively and even aerial leaping. It is common to have females end up out of the pond when their aerobatics take them in the wrong direction. Eventually, the female will succumb and the pair will go about looking for the ideal spot to lay the eggs.

The wild antics are usually quick obvious. At the first sign of spawning, the pond owner should be thinking about whether they want to save the eggs or let Mother Nature take her normal course.

Like us, all occupants of the pond find eggs to be delicious, nourishing food. Any eggs laid in the pond can quickly become a meal. Eggs can be spread over quite a large area if the fish spawn in natural plantings. If there is plenty of cover, baby fish will hide and a few will survive. In the open, they have little chance of survival. Fish breeders collect, then separate the tiny babies to increase their chances. Even the best techniques will leave an owner with only 10% of the babies after the first year. But, think about the fact that an adult Koi can lay 5000 to 50000 eggs. That means that after the first year, a breeder will still have 500 or more babies, and possibly thousands.

A neat trick when dealing with spawning fish is to use a fabric spawning basket.
This is a fuzzy fabric based, bucket shaped, floating basket that hangs from the surface of the pond.
Fish find this to be an ideal surface upon which to lay their eggs. The very fine sheepswool-like fabric causes the fish eggs to stick well and the pile type material can hold thousands of eggs on it's outer surface. A female fish will circle the basket, rolling up on her side and squirting out the eggs onto the fabric surface. The male will be close behind, releasing sperm into the water which will fertilize the eggs. When all of the commotion is over, the breeder will reach in and grab the bottom of the basket. The basket is slowly and carefully inverted and set back into the pond. The eggs are now inside of the basket where they are safe from predation. The eggs hatch in 2-3 days, so now is the time to prepare a growth tank for the little guys. A breeder would already have a tank prepared and waiting.

The growth tank should be 100-500 gallons in size and have a dense amount of submerged plants to give the babies hiding areas. The water should be seasoned pond water and definately NOT water that had any amount of chemicals in it within the last several weeks.  We prefer a salt level that is considerably lower that of our big ponds and test for a level around .001%. Ammonia level should not even be measurable. An old sand or heavy clay bottom is best, and a dense planting of Anacharis or other bushy submerged plant is best. You should have newborn food available, and the tank should NOT have any adult Mosquito fish in it. Newborn fish are very fragile. Cleanliness and excellent oxygen levels are extremely important.

This next paragraph will explain in detail why there are so few fish that survive the first year.

Some of the newborn fish will grow at twice the rate of others. Within weeks, you will see baby fish ranging in size from 1/4" to 3/4" in length. The larger babies will eat the tiny ones. The denser is the cover, the better the little guys can hide. Green water is fine, the suspended algae that makes water green is actually healthy for small fry, and the lower visibility makes predation more difficult.

We have also used screens inserted vertically in the tank every foot or so. The material used should have a grid sized about like a wire collander. Speaking of which, we used stainless steel dollar store collanders hanging from rods one season. There is no way to get real numbers about what works best so we just try everything to give the small fry as much chance of survival as possible.

By the time the smallest fish reach an inch or so in length, the cannibalism stops as they are too big to swallow and Koi and Goldfish don't have teeth capable of chewing large items.

Feed very fine food in small amounts in the beginning. Quality Koi food can be ground in an old coffee grinder to make a fine powder for the babies.

Some points to remember;

Make a fabric screen for any pump used in a growth tank. Pumps do eat tiny fish.
Check ammonia levels frequently. You want ZERO ammonia.
Keep air pumping in the tank for good oxygen levels.
Shade the tank from more than 3-4 hours of direct sun. You need that much for the plants.