Though more important for fish who obtain all of their required oxygen from the water column via their gills (as opposed to bettas and other labyrinth fish who supplement this with atmospheric oxygen that they gulp from the water's surface), the amount of dissolved oxygen in the water column is important to all fish and should be considered for your betta.
The amount of dissolved oxygen (and other gasses) in water is influenced by two factors:
Because you cannot create a pressurized environment in your aquarium, the only factor that you can influence is surface area. Simply put, the larger the surface area relative to the water volume, the greater the oxygen dissolution will be.
Given this, you can see that a tall and narrow tank will have less oxygen dissolution than a long, wide tank of the same volume. This should be kept in mind when choosing a tank shape for your fish, as the shape is not only for aesthetic appeal - it is very functional as well.
Given the above, the surface area of the water in your tank is important to bettas for two reasons:
To best facilitate the above, you should consider two aspects of your tank's dimensions:
You may often see it suggested that you add an air stone to your tank to increase aeration. We would like to clarify the manner in which air stones help to bring more oxygen into aquarium water.
One may see the bubbles and assume that by adding bubbles to your tank, you are adding oxygen to it. It is not the bubbles in the water, but in fact the bubbles as they reach the water's surface that increases oxygenation. Since you cannot pressurize your tank, none of the bubbles that travel through the water will add any oxygen to your tank. However, when the bubbles break the surface of the water, they cause it to bulge upwards and ripple as they pass through, effectively increasing the water's surface area. As mentioned above, increased surface area leads to increased oxygen dissolution into your tank, and it is only in this way that bubblers help to increase oxygenation.
The same effect can be accomplished, therefore, by causing any disturbance of the water's surface, which will change/bend and warp it's shape, increasing the surface area. Increasing the flow of your filter, for example, would increase water surface agitation, effectively increasing surface area and thus oxygen dissolution.