Evasion is a class feature gained by Rogues and Monks at level 7:
At 7th level, your instinctive agility lets you dodge out of the way of certain area effects, such as a blue dragon’s lightning breath or a Fireball spell. When you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you instead take no damage if you succeed on the saving throw, and only half damage if you fail.
Does this feature allow a character to “dodge” area effects, taking half damage, even if they are unconscious?
Yes. RAW, Evasion means a character only takes half damage, even when unconscious.
The relevant part of Evasion text, that you’ve quoted says:
When you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you instead take no damage if you succeed on the saving throw, and only half damage if you fail. (SRD p. 28, 38, 40)
The Unconscious condition states:
An unconscious creature is incapacitated (SRD p. 359)
Which means they:
can’t take actions or reactions. (SRD p. 358)
However, nothing in the ability Evasion says that it requires an action or reaction for use. So Evasion is an ‘always on’ ability.
The triggering condition is being ‘subjected to an effect that allows you to make a Dexterity saving throw’, not whether or not the creature is actaully able to make that throw, due to other conditions already affecting them. If this caveat was intended it should be explicit (in 5e ‘there aren’t any secret rules’).
However, also while Unconscious:
The creature automatically fails Strength and Dexterity saving throws. (SRD p. 359)
So, while a character is unconscious they’ll automatically fail Dexterity saving throws, but thanks to Evasion they’ll still only take half damage.
If this ruling seems odd to you (and I can see why it might), then you are, of course, always welcome to rule otherwise in your own game.
But, mitigating circumstances mean that this isn’t as bad as it seems…
Firstly, when a character is already unconscious due to HP loss, they are less likely to be targeted by AOE attacks (so this will be an issue less frequently than you might imagine).
But secondly, and much more importantly (presuming that the target isn’t simply asleep or drugged, and with the exception of the ‘massive damage’ rule), how much damage you take whilst unconscious doesn’t normally matter at all.
What matters is how many death saves you’ve failed.
If you take any damage while you have 0 hit points, you suffer a death saving throw failure. (SRD p. 98)
Taking any damage is always going to equate to one failed death save, whether or not it’s halved. (Critical hits would cause two failed death saves, but non-attacks such as spells and other abilities that force you to make a Dexterity save can’t be critical hits anyway.)
So, Evasion will not help a character to take less failed death saves.