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The first part of the description for Blink reads:
Roll a d20 at the end of each of your turns for the duration of the spell. On a roll of 11 or higher, you vanish from your current plane of existence and appear in the Ethereal Plane (the spell fails and the casting is wasted if you were already on that plane)....
If our wizard cast Blink at the start of an encounter and was then knocked unconscious, would she still go to the ethereal plane as part of the effect of the Blink spell?
The spell Blink doesn't require Concentration, so I'd say it carries on working regardless of what happens to the caster after it has been cast.
Spells which have very long durations (such as Geas) continue to work even when the caster is asleep, for example.
Blink also states:
At the start of your next turn, and when the spell ends if you are on the Ethereal Place, you return to an unoccupied space of your choice that you can see within 10 feet of the space you vanished from. If no unoccupied space is available within that range, you appear in the nearest unoccupied space (chosen at random if more than one space is equally near).
I would say that the "nearest or random space rule" applies, as the caster wouldn't be able to chose a space to reappear in when unconscious.
It's not a concentration spell. If it were, the answer would be clear, since with concentration spells, the PHB says "You lose concentration on a spell if you are incapacitated". But blink isn't a concentration spell. The only thing helpful in the spell is that it says, "You can dismiss this spell as an action".
Based on all of this, my interpretation is that RAW unless the caster dismisses it with an action, then the caster keeps blinking for the full minute whether you're conscious or not. Of course, something else could interfere with it, such as someone casting dispel magic, but that's out of the scope of the question.
One issue is that the spell says the caster returns to an unoccupied space of your choice that you can see. Clearly the unconscious caster can neither choose nor see. The GM will have to adjudicate that. I would interpret it that the caster blinks back to the same place, unless it becomes occupied, then they blink back to a nearby place.
