I beta for several people, and it is up to them to say whether I am a good beta or a bad beta. That said, this is what I do:
1 -- Is the story worth telling. This is a hard thing to discuss. Publishers will determine in a first page if they continue. So a Beta must be able to be honest enough to say, I don't think this will grab people enough to be publishable. Then a beta has to go the next distance and tell the author why. It isn't enough to say: I don't have an emotional connect with this character. You must go the next step and say why you don't, and then suggest how it might be remedied.
2 -- Fact checking. A story based upon a false premise can not ring true. That said, a good Beta also should check to see if there really is a Rylance Drug Corp in Sheboygan, and if so, the author has to consider changing the name vs being sued for saying they are selling poisons.
3 -- Spell, syntax, punctuation. Stories which flip tenses, are full of misspellings run on sentences and poor structure will be circular filed by the publisher. The author should make a best attempt to present a beta with these in fine form. A beta will ferret out where the author missed a beat, and point it out.
4 -- I trouble. The author is not the character, When the author is too in love with the character and when that shows, it is a sign of I trouble. Betas should point this out. (See: Mary Sue)
5 -- Redundancy redundancy redundancy. The beta should be certain to let the author know NOT to use the same word over and over and over and -- oh yeah -- over. If I am distracted by seeing the same word at the beginning of every sentence in a paragraph, or every paragraph on a page? Let the author know. Editors and publishers will be less... indulgent.
6 --
You keep using that word. Sometimes even the best of us gaff on a word. A beta can suggest a new word. A better word.
7 -- Betas do not rewrite in their own image. NO NO NO! The beta's duty instead is to get the Author to hone what s/he is writing in the manner that they are writing, Beta's don't recut the diamond, they polish the diamond so it shines.
And that, in my humble opinion is what a beta does.