We cover this in some detail in other places on this website but seeing as how it is such an important topic you need to read this page carefully.
Aeration in a Koi pond or a body of water is poorly understood. But what helps is to understand that there is only one place where oxygen is able to get into the Koi pond in any meaningful quantity. This is at the surface of the pond where the pond meets our atmosphere.
In other words, waterfalls and venturis DO NOT oxygenate Koi ponds at all. In terms of the quantity of oxygen that they add you can safely work on the assumption that it is zero.
Venturi's do not oxygenate the water. What they do is that they add bulk to the returning water stream from a pump by adding air bubbles and hence volume into the water stream. This then increases the movement of the water stream in the pond because of the increased volume of the combination of air and water.
In other words the pump is able to push more water around in the pond with a venturi than without it. That's it. Period.
The bubbles from the venturi are simply not in contact with the water for enough time for any meaningful amount of oxygen to transfer from the bubble to the water body.
Similarly waterfalls have little meaningful oxygenation contribution. Unless the waterfall is miles long the time the water spends on the actual surface of the water fall is not sufficient to ensure that this water is thoroughly oxygenated by the time it hits the surface of the Koi pond.
What actually does happen in oxygenation is that water at the surface of the Koi pond exchanges gases with the atmosphere in the same way that our lungs exchange air from our blood with the atmosphere.
Carbon dioxide is released into the atmosphere and oxygen is dissolved.
Waterfalls operate in ... the surface of the Koi pond. So do venturis.
It stands to reason that the water that has the least oxygen in a Koi pond is at the bottom, where it is furthest away from the surface. And that the most oxygenated water in a Koi pond is at the surface. This explains why fish under oxygen stress gasp at the surface of the Koi pond.
Therefore, in order to ensure that Koi ponds carry the maximum amount of oxygen that they can, it is necessary to move the oxygen poor water from the bottom to the surface of the pond. This allows oxygen to dissolve into the Koi pond water.
This type of circulation is referred to as vertical circulation. Water is moved vertically from the bottom of the pond to the surface. If this process is continuous you will find that the oxygen levels at all points in the Koi pond are exactly the same, and ideally the water is completely oxygen saturated.
Having all the water in your Koi pond oxygen saturated is of immense benefit. It means your filters are being fed oxygen rich water. The Koi are subject to oxygen rich water at all points and Koi hobbyists that want their Koi to grow know for certain sure that having high oxygen levels in a Koi pond result in Koi that grow faster, grow bigger and are healthier than Koi in ponds where oxygen levels are not as high.
Achieving vertical circulation in a Koi pond is simplicity itself. Airstones located near or at the bottom of the Koi pond deliver a fine mass of air bubbles which rise up through the Koi pond water body. As they do these bubble pull water up with them, to the surface. A single air stone can deliver thousands of litres per hour of water from the bottom of the pond to the surface - and consequently can elevate the oxygen levels in the pond massively.
There is no downside. And hence an air pump is an almost mandatory piece of equipment on a Koi pond. It is and remains, the single biggest immediate improvement you can make to your Koi pond if you don't already have one.