How do you want your spells to work?
In the PHB section on Spellcasting, we see that:
A typical spell requires you to pick one or more targets to be affected by the spell’s magic. A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area of effect...
Your initial assumption is:
I know that spells, in general, state what they can target - which is often only creatures...do spells that usually target creatures have the ability to harm the Prismatic Wall spell?
It seems to me that you are conflating the issue of choosing a spell target (what must be present for the spell to be cast) with resolving what classes of things a spell may legitimately affect after it is cast. Just because a spell requires you to choose 'a target to be affected by the spell's magic' does not mean that this target is the only thing that the spell affects or can harm.
In particular, for 'area of effect' spells (like cone of cold), the target you choose is the spell's point of origin; after this, the spell can then affect things within its 'area of effect' without you having to choose them and without them being specified as targets. These other things might be called 'targets' in the spell description, or they might not. For example, fireball explicitly says that it can damage creatures, and calls them targets, but it also explicitly says that it ignites flammable objects, while not calling them targets. Flame Strike, on the other hand, explicitly damages creatures, but the creatures are not called targets.
As above, "A spell's description tells you whether the spell targets...a point of origin for an area of effect". For an area of effect spell, the only target you choose is the point of origin; for what the spell affects you need to read the spell description.
Cone of Cold says:
A blast of cold air erupts from your hands. Each creature in a 60-foot cone must make a Constitution saving throw. A creature takes 8d8 cold damage on a failed save, or half as much damage on a successful one.
We know that this is an area of effect spell, so the target you choose is the point of origin of the cone. For what else can be affected by the spell, we need to read the spell description. We can see that the spell explicitly affects creatures, but could it affect other things?
Here you have a fundamental choice to make - are you reading the spell description in a prescriptive way or a descriptive way? A prescriptive reading would take the spell description to mean that it does exactly and only what it says it does. A descriptive reading would use the 'no flavor in spells' principle to construct other effects not explicitly spelled out so long as they were not excluded by the spell description or other rules, following the principle of 'the DM describes the results of your actions'.
Both of these approaches are RAW, and both have their advantages and disadvantages. The prescriptive approach has the advantage of being very clear in interpretation and more uniformly applied across DMs, which is important in situations like Adventure League play. Unfortunately, in cases like this (can a cone of cold damage the red layer of prismatic wall?) a prescriptive reading produces the result of 'the rules are stupid, so ignore them'. A prescriptive reading admits that yes, the spell produces a blast of air that is objectively cold, but then says that this blast has no further effects because the spell does not explicitly state them.
A descriptive reading, on the other hand, is more subjective. Since the spell says that it produces a 'blast of cold air', each DM will have to determine what the subsequent effects of this blast are, because it is just as much an effect of the spell as the explicit damage to creatures is. However, the descriptive approach has the advantage of using RAW to get the much more satisfying result of the cone being able to take down the wall.
How cold is this blast of cold air? It is so cold that it does 8d8 cold damage. What kinds of things would this damage? Anything in its area of effect that is capable of being damaged by cold. We know that it damages creatures, because the spell description says that it damages creatures, but it does not say it damages only creatures, so with a descriptive approach we are free to consider other possibilities.
Can this cold damage objects? No, because objects are immune to effects that cause Constitution saves (as per the general rule on PHB 185).
Can this blast of cold air cause environmental effects? Could it frost over the surface of a water trough, cause condensation droplets to form on beer steins, or blow loose papers off a desk? A descriptive reading would say sure, the DM describes the results of the players' actions and any of these effects are fair game, so long as they don't rise to the level of 'causing damage to an object' which is prohibited as above. A prescriptive reading would say that none of these effects are possible because they aren't explicitly permitted by the spell.
The red wall of Prismatic Wall is neither a creature, nor an object; it is a spell effect. Spell effects are special, and in their descriptions they state what they can be damaged by, if anything. A spell effect that does not specifically say it can be damaged by cold would not be damaged, but a spell effect that says it can be damaged by cold will be.
Prismatic Wall says:
Red: ...The layer can be destroyed by dealing at least 25 cold damage to it.
When the cone of cold intersects with the wall, we have a spell effect whose description says it does 8d8 cold damage overlapping in space with a spell effect that says it is destroyed by 25 points of cold damage. Because the wall is not a creature, it does not get a save. Because the wall is not an object, it is not immune. Because the wall says it can be damaged by cold, it can be. 96% of the time, the wall will be brought down by the first cone of cold cast.
Thus, a descriptive reading of the spell gives us the answer that we intuitively want anyway, and still supports the RAW framework of the game. I think that is pretty useful.