Which one of the following sentences is correct:
"How Internet helps you?"
"How does Internet help you?"
Is it always mandatory to use "do" in how-questions?
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OP's example Internet (without the definite article) is a distracting non-standard usage. See “I don't have internet” vs. “I don't have the internet” as previously asked on ELU for further discussion of this).
So to focus on the specific aspect of usage being asked about here (using "do" to frame a question), let's change it to...
1: How Facebook helps you
2: How does Facebook help you?
Notice that in my amended version, I've removed the question mark from #1. That's because it's not a question. It's not even a sentence - it's just a noun phrase. We can see this by considering...
Quantum physics is a complex subject (Noun Phrase + Verb + Noun Phrase)
He explains quantum physics (Pronoun + Verb + Noun Phrase)
He explains how Facebook helps you (Pronoun + Verb + Noun Phrase)
As to whether it's necessary to include do in questions, consider this answer to the earlier ELL question “How it works?” vs. “How does it work?”. Effectively it's not, since these are all valid questions...
How can Facebook help you?
How will Facebook help you?
How could Facebook help you?
How has Facebook helped you?
How is Facebook helping you?
That's to say, the interrogative (how, in this case) must be followed by an auxiliary verb, but there are many others besides to do that can perform this function in questions.
Of the two examples you quote:
How the Internet helps you?
A2) How does the Internet help you? (subject-operator inversion)
only A2 is correct. Questions like A2 require what is known as a subject-operator inversion. In this case the subject is the Internet and the operator is does. It is called an inversion, because in English a subject usually precedes a verb.
Although subject-operator inversions are frequent in interrogative sentences, there are cases where this inversion doesn't happen. For example:
Wh-questions where the wh-word is the subject
Who are you?
What is causing that racket?
Declarative questions are questions used to ask for a confirmation, e.g.:
A: She quit her job?
B: Yes, she did! I couldn't believe it either.
Dependent interrogative clauses It is possible to rephrase a question so that it becomes a dependent clause. When this happens the subject-operator inversion disappears:
I'm asking how the Internet helps you. Could you explain it to me?
"How does the Internet help you?"
As there is only one Internet, you need the definite article "the" in front of it.
You do not need to always use "do" with a "how" question. You could, for example, ask, "How is the Internet helpful to you?" or "How can the Internet help you?" or "How long has the Internet existed?" "How do/does" questions ask in what manner one performs some task or the manner in which something is affected. I'm not sure that I could catalog all the types of things you could ask with "how", but that's only one of them.