Which of the following is correct?
Both of you know about it.
or
Both of you knows about it.
Which of the following is correct?
Both of you know about it.
or
Both of you knows about it.
The "s" at the end of that verb is called a third person singular "s", and as you can guess from the title, it's added to third person singular verbs only, which would be "he", "she", and "it", or any other noun that can be exchanged with either of those subject pronouns.
The subject in your sentence doesn't fit that category, 1. because it's not singular, 2. because it's not third person, so you don't need the "s".
Both of you know about it.
Both is the subject of your sentence, and it is what the verb should agree with. Since Both is plural, you should use the plural form know, not the third person singular knows.
Both of you know about it.
of you modifies Both, and does not affect the verb's number.