The Indigo layer of a Prismatic Wall states:

While this layer is in place, spells can't be cast through the wall

By comparison, a Globe of Invulnerability states:

Any spell of 5th level or lower cast from outside the barrier can't affect creatures or objects within it, even if the spell is cast using a higher level spell slot. Such a spell can target creatures and objects within the barrier, but the spell has no effect on them. Similarly, the area within the barrier is excluded from the areas affected by such spells.

It is quite clear that, say, a Fireball cast outside of a Globe of Invulnerability can’t extend inside it... But Prismatic Wall seems to lack this clause. So, would a Fireball, or other area of effect spell, cast on a point on one side of a Prismatic Wall extend through the other side of said wall?

Yes, area effects will cross through the walls

There is no language in any of the layers, or the general wall itself, that state it acts as cover or that spell effects can't pass through it.

Prismatic Wall is described as(emphasis mine):

A shimmering, multicolored plane of light forms a vertical opaque wall--up to 90 feet long, 30 feet high, and 1 inch thick--centered on a point you can see within range.

The wall itself is only light, with each layer providing a specific effect. An area of effect spell, such as fireball is not physically obstructed from passing through, so it can pass through.

Casting Through vs Effects passing through

The only relevant wall type is violet, which states:

While this layer is in place, spells can't be cast through the wall.

The limitation here is being able to cast through the wall. In this case, the spell isn't being cast through it. If the point of origin is outside the wall, then you are not casting through the wall. Your point of origin is outside, so you can cast. The spell effects can pass through because there is no limitation on that, only casting through it. The language from globe of invulnerability covers this specific case - and because the violet wall doesn't have that language, then creatures behind the wall can still be affected by a spell that is cast on the other side.

The creatures under the effects of the spell just happened to be within range of your point of origin. If your point, and therefore your spellcasting, doesn't cross the violet wall, then targets within range of that point but on the other side of the wall would still be effected.

A spell is only being cast through if the spellcasting portion of it crosses. For Area of Effect spells, like fireball, that is not happening. The effects of the spell cross over, but the spell is not being cast through the barrier.

Yes, AOE effects would reach through

spells can't be cast through the wall

Casting a spell through the wall would mean casting on one side with the target(s) on the other side. However, AOEs generally target a point of origin:

A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area of effect.

If whatever the target of your spell is is on your side of the wall, then you are not "casting through" the wall and the spell works as the description indicates.

The only other things that would affect an AOE from affecting though the wall would be specific wording in the spell preventing it (which, as you noted, it doesn't have) or if it granted full cover and thus prevented spells and effects from moving through it. The wall does not provide full cover being that is is made of light and that things can move freely through it.

A shimmering, multicolored plane of light forms a vertical opaque wall [...]

Thus, there is nothing that would prevent an AOE's effects from traveling through the wall as long as the target was on the same side of it as you.