Who is responsible for leaving the window open?
This is completely correct. It is somewhat high register, and would normally be interpreted as a question of who should receive blame for the offense of not closing the window when it should have been closed.
The other option is not correct, but there’s actually a lot going on there. Let’s go piece by piece:
“Responsible” can also be used in positive contexts, to describe someone charged with a certain duty. Because the “act” of leaving the window open is really a sin of not acting, it is fairly clear what is meant here, but there are cases where a different verb might make this construction less clear. For example:
Who is responsible for cutting the cake?
This would be a valid question in cases where the cake was not supposed to be cut (and someone cut it anyway), but the default interpretation would be that this is a question about who should cut the cake.
“The responsible” could be used to collectively label a group of people who would each be described by this adjective. This is a fairly specific, advanced usage, but is similar to “the good, the bad, and the ugly.” It doesn’t really make sense to use it here, because the rest of the sentence is formed in reference to a single person and not a generalized group and you don’t mention a specific entity for “responsible” to describe as an adjective. A possible correction of this could come from adding mention of a “person”:
- Who is the person responsible...
- Who is the responsible person...
Be careful with option #2, though. That order makes it sound like you are describing the person as responsible in general (positive characteristic) and runs the risk of making this a garden path sentence where the audience has to mentally go back and re-interpret “responsible” as an indication of blame. “Who” is allowed to be general, but if you add “the” you’ll have to specify.
Let it or leave it
Similar to “let it be”, there are regional, colloquial uses of “let it” that would connote something similar or identical to “leave it”. As mentioned above, the rest of the phrase is fairly formal so it would be a jarring mismatch of registers within the context of this here collocation (see what I did there?).
Outside these regions and registers (and the rest of the sentence currently does pull us out of casual register), the more formal or literal definition of “let” takes over.
If someone “leaves” a window open, it means that—despite opportunity and/or obligation to close the window—they failed to do so. The usual interpretation of “let”, on the other hand, would suggest that the window is going to open unless someone intervenes. If the situation is that a faulty spring causes the window to open unless someone prevents it, “let” might be the more appropriate word for the question. It’s still a question of inaction in either case, but the implication is either simple negligence or failure to sufficiently counteract a force that would open the window.
As Jay notes in his sage comment, the choice between “leave” and “let” has an effect on whether “open” is an adjective or a verb. When a window is left open, “open” is the state of the window; when it is let open, it is allowed to open.
This is why your choice of “opened” doesn’t fit. “Let” makes the reader expect a verb, and as a verb “open” is secondary to “let” and should be in its base form. “Opened” works fine as an adjective and could be substituted into the correct version of the sentence without making it ungrammatical, but most people would probably just say “open” anyway.
Possibly the primary problem with the “let” phrasing is that it does not employ the gerund (as your selection of the “gerund” tag indicates you are aware). What’s happening here is a gerund clause used as subject.
Note that this is all only necessary because “is” locks you into a present reference-frame. The simpler question would be:
Who left the window open?