Identify the part of speech for the word "BOTH" in a sentence

Both the brothers are actors.

"Both" is neither an adjective nor a pronoun. It belongs to the word category (part of speech) 'determinative'.

Its function here is that of 'predeterminer' (modifier), a type of quantificational expression.

Although "both" is located within the NP, it is outside the head NP "the brothers":

[Both [the brothers]] are actors.

Outer brackets enclose the NP, while the inner brackets enclose the head NP, which is modified by "both".

Note that the determinative "both" can also function as an adjunct, as in:

We both enjoyed it

Here, unlike predeterminer "both", the adjunct "both" is not part of the subject NP, but an optional quantificational expression in clause structure.

This is a tricky question. The online Cambridge English Dictionary lists both as:

predeterminer, determiner, pronoun

So all three, depending on the context. This online version goes on to provide a lengthy series of illustrations of the uses of each status.

It is worth looking at what this word is doing in its differing uses, in a language that has different words for the different functions.

Tous les deux frères sont des acteurs (French)

Kai oi duo adelphoi einai ithopoioi [Καί οι δύο αδελφοί είναι ηθοποιοί] (Greek)

So neither language uses a special word to do the job of English both here. French uses a phrase meaning all the two, which is what we do when the number rises to three: in English we then say all three brothers are actors. In this we are like the germanic languages. German works in the same way:

Beide Brüder sind schauspieler, but alle drei Brüder sind schauspieler (all three brothers are showplayers!) - why wouldn't English follow the German?

Greek uses a conjunction to solve the problem (literally 'and the two'). They do the same for 'both ... and'

Kai su kai ego ithopoioi eimastε [καί σύ καί εγώ ηθοποιοί είμαστε]

This Greek usage of an initial 'and' (which comes down from ancient Greek) adds nothing at all to the meaning of the sentence, any more than the 'Both you and I are actors' alters the truth conditions from those of the sentence without the 'both'. It is a little like a vital feature of the ancient language, the particle. It adds colour or gives a warning about what is coming later. In a way some of them could be called 'pre-conjunctions' or 'premonitives' (pre-warners). So 'te (τε)' after the first word of a sentence says "look out, at the end of this statement, another one is going to be added". And sure enough, and the end of that statement, arrives the reassuring 'kai' and we know where we are. If, instead, the second word is 'men(μεν)', we are being alerted to the fact that the first statement is going to be followed by something that contrasts with or contradicts it. As before, we know we have reached that point when we see/hear the enclitic particle 'de (δε)'. In English we have a slightly clumsier way of achieving the latter effect by the phrases 'on the one hand ... on the on the other hand.'

There were more of these guiding and shaping the reader's and (more often) listener's journey through the speech, lecture or recital: 'ara(αρα)', as first word = 'here comes a question'; 'ar'ou(αρ'ου)' = 'here comes a question to which I think the answer is yes'; 'ara me(αρα μη)' warns that I think the answer is no. The same particle can appear all over a sentence colouring our intended reaction as one of scepticism, surprise, scorn. The now much cursed word 'so' functions a little like that, perhaps, though with different purposes.

'Both' is also quite like the particle, when used in the sentence:-

I can either stand on one leg with my eyes shut or have a blazing row with you, but not both together.

So what is the answer to your question? 'Predeterminer' is the probably best, given that the choice here is limited. I recommend a good look through the Cambridge Dictionary entries on the words both, 'determiner' and 'predeterminer', where they provide a lot of useful examples.