First, write the sentence in the positive, and see whether the verb that you want to negate is the main verb in the sentence. Consider talk in these three sentences:
he talks a lot - talk is main verb
he really likes to talk - talk is not the main verb
he likes talking - talk is not the main verb
Main verbs
If you want to negate the main verb and there is already an auxiliary verb (be, have, will, should could, must) in front of it, you insert not between the auxiliary and main verbs.
I am not going to the party
I have not seen him today
I will not have time tomorrow
You must not tell anybody about this
If the verb that you want to negate is be, just add not after it.
Dinner is ready
Dinner is not ready
The main verbs need and dare are special cases: in the positive, it is followed by to, and in the negative to is replaced by not.
You need to come
You need not come
For all other main verbs that don't have an auxiliary verb, you add the auxiliary verb do and then negate the auxiliary verb. The auxiliary verb has two cases: does for third person singular (he/she/it) and do for everything else. When negated, this can be written out in full do not, does not, or contracted don't, doesn't.
I want to go to the party
I don't want to go to the party
non-main verbs
If the verb that you want to negate is not the main verb and it's preceded by to, it's an infinitive: you negate it by putting not before the to
I told him to come
I told him not to come
Likewise, if the verb you want to negate is not the main verb and it's got an '-ing' ending, then you negate it by putting not before the verb.
thinking can get you into trouble
not thinking can get you into trouble
"How dare I!" would be an ironic (or sarcastic) play off of the common expression "How dare you!" used to suggest that the offense you caused was unreasonable or even ridiculous.
For example, suppose my friend is in a bad mood, and she gets angry when I cheerily greet her and compliment her:
Oh wow. I'm such a jerk. First I had the audacity to wish you good morning, and then I made it worse by saying you look really nice today. How dare I!
Otherwise, if you are seriously apologizing for some injury, then there many other expressions you can use:
I'm really sorry that I hurt you.
It's all my fault that I hurt you.
It's terrible what I did to hurt you.
I'm so ashamed of how I hurt you.
I really messed up. I didn't mean to hurt you.
And so on.
(Edit) As JBH's comment says, it's possible to use "do I dare" or "dare I ..?" as a rhetorical question, but this is not suggesting fault as much as questioning courage or capacity.
Alex: That attractive woman in the cafeteria has been sitting alone every day this week. Dare I go up and ask if she would like some company?
Bob: Go for it, man!
Best Answer
The grammar pattern don't you try to do something, which is used quite widely in English and of which the expression don't you dare do something is just a variant, is an emphatic way to say don't even think about trying to do that where you're placing extra stress on the fact that it's probably not going to be very wise of you if you try to do whatever it is that you're planing on doing. Whereas don't try to do something sounds simply more like a command or request urging you not to do something. Take a look at these examples:
Here, I'm telling you that you should not even think about being worried that I will be late delivering the books to you. There is no point in being worried that you're not going to get them. In other words, rest assured that you absolutely will get them.
This one sounds exactly the same as the previous example with the exception that it sounds a little less emphatic. That's really all the difference there is between these two examples.