Does it matter whether the interval is diminished, minor, major or augmented in transposing THE KEY SIGNATURE? of course it is crucial in the transposition of notes but is it important in key signatures and if so how? I have been told that my question is not clear enough so here is my example: For example, B major down a minor 3rd
I think the answer is "yes". (In any theory question, the answer to "Does it matter?" is almost always yes.)
Your example -- "down a minor 3rd from B major" -- must mean changing to a key with the note name two lower, i.e. G; and to make this a minor third it must be G# major. The similar question "Transpose down an augmented 2nd from B major" results in the same notes on the piano, but goes to the key of A♭ major.
Was that the question? Actually, perhaps you thought up the example yourself, because it is not a good one: it is not normally possible to write the key signature of G# major, since it has eight sharps, and thus includes a double-sharp.
A key is definitive. So key signatures are too. Changing the key of a piece (transposition) necessitates changing the key signature. It matters not what intervals are included in the original - they will end up as the same intervals in the transposed music.
For example: in key C major, notes E and G make m3. Change the key to D major, those notes will become F♯ and A. Still m3. The fact that D has F♯ in the key sig. does it all automatically. Were there say, B♭ and C (in key C), that would still be M2 in key D. And the notes would need accidentals, so in key D, they'd be C♮ and D. Still M2.
...If that's what you're asking, as the question is not very clear.
Are you asking whether an exam question 'transpose this passage up a major 3rd' requires you to change the key signature? It's a fair question. In 'real music' a musical phrase may well occur transposed to several keys, and we generally don't keep changing key signature. But I think the sort of transposition exercise you'll find in a grade 5 theory paper WILL expect the key signature to be transposed as well as the notes.