For example, a 12th level character produces an effect that has "x'/level" area of effect. Can the character control the size of the area of effect, or not? It would be nice to have a quote of where it is specified for powers (and spells), if possible.
You can cast a spell at a lower caster level than normal, but the caster level you choose must be high enough for you to cast the spell in question, and all level-dependent features must be based on the same caster level.
You can manifest a power at a lower manifester level than normal, but the manifester level must be high enough for you to manifest the power in question, and all level-dependent features must be based on the same manifester level.
So by default you can reduce the area of the spell or power, but not without also reducing its other level-scaling parameters. Particular abilities (Sculpt Spell/Power metamagic/psionic feats, the archmage’s mastery of shaping high arcana/archpsion’s sculpt power high psionics, etc.) can create exceptions to this, however.
A typical manifester generally can't opt to have a power affect a littler area
No general rule says that a manifester can change a power's listed area. However, manifesting a power at a lower manifester level can change a power's area if the power's area depends upon the power's manifester level. That is, on Manifester Level says
You can manifest a power at a lower manifester level than normal, but the manifester level must be high enough for you to manifest the power in question, and all level-dependent features must be based on the same manifester level. (Expanded Psionics Handbook 54)
For example, a psion 15 could manifest the 7th-level psion/wilder power divert teleport [psychoportation] (96) at a manifester level of anywhere from 13 to 15 but not lower than 13, and the power's area would be altered in proportion to the picked manifester level.
A typical caster suffers a similar restriction on a spell
Likewise, no general rule says that a caster can whimsically change a spell's area. Further, rules nearly identical to those for powers apply to spells (i.e. "You can cast a spell at a lower caster level than normal, but the caster level you choose must be high enough for you to cast the spell in question, and all level-dependent features must be based on the same caster level" (Player's Handbook 171) here)).
For example, a Wiz13 could cast the 6th-level Sor/Wiz spell control water [trans] (214) at any caster level from 11 to 13 but not lower than 11, and the spell's area would be altered in proportion to the picked caster level.
A specific character can devote resources like class levels, feats, and gp (in the form of magic and psionic items) to change a power or spell's area—some examples of which can be found in fine answers here and here—, but changing a spell's area is not an option generally available to all casters or manifesters for free.
YES by reducing your caster level and with the Sculpt Spell Metamagic feat but what you can is limited
(There was no psionic tags when I first answered this question as far as I remember)
As both KRyan and Hey I Can Chan pointed out you can indeed reduce your caster level to do so and they also covered the psionic/powers part that was not included in the original question KRyan also included more ways to modify your spell area.
Another way would be to use the Sculpt Spell Metamagic feat wich states:
(Complete Arcane p.83)
You can modify an area spell by changing the area's shape to either a cylinder (10-foot radius, 30 feet high), a 40-foot cone, four 10-foot cubes, a ball (20-foot-radius spread), or a 120-foot line. The sculpted spell works normally in all respects except for its shape. For example, a lightning bolt whose area is changed to a ball deals the same amount of damage, but affects a 20-foot-radius spread.
A sculpted spell uses a spell slot one level higher than the spell's actual level.
So it is useful to modify the spell area and it could reduce the surface of the spell by increasing it's height instead if you change it to a cylinder for example.
The ''four 10-foot cubes'' is an interesting choice, you can target 4 specific areas (within the spell range limit of course) and they don't need to be adjacent.
It does mention that you modify the area of the spell so a spell without an area of effect cannot be modified by the feat, lightning bolt has an area of a 120ft line (given example) so it can be modified.
Spells with area different from the options of Sculpt spell such as control water could be modified since they have an area but I guess a discussion with the DM would be a good idea.
- Sculpt Spell has specific boundaries that constrain the spell area