The correct option is
A Had I known about this, I would not have allowed you to play with those kids.
'If' is a conjunction which is used in a present or future conditional sentence to express uncertainty about whether an action (or event or condition) will occur. In the event, the particular action (or event or condition) occurs, then the other event is likely to occur.
Option A: 'Whether' is a conjunction which is used to express a choice between two alternatives or events. Here, there is no such reference. Hence, this option is incorrect.
Option B: The auxiliary verb, 'had' can be used to replace the word' if' in a conditional sentence in the past tense. The given sentence is in the past tense. Hence, 'had' can be used. The auxiliary verb 'had' is used along with the past participle form of the main verb. The main verb is 'knew' whose past participle form is 'known'. So, the clause, 'If I knew about this' can be replaced by 'Had I known about this'. Hence, option B is correct.
Option C: A conditional sentence can begin with the auxiliary, 'were' when the original sentence (starting with 'If') has 'were' as the linking verb (which acts as the main verb). In the given sentence, 'were' does not appear. Hence, this option is incorrect.
Option D is incorrect as a conditional sentence cannot start with the words, 'Have I'.