1. She is taller than I/me (am)
2. You know much better than I/me (do).
Some people feel that the sense requires the bracketed verb I have added, and that therefore the subject pronoun 'I' is required.
Others feel that 'than' is functioning as a preposition, that we don't need the bracketed verb, and that the object preposition 'me is correct.
In BrE, you will probably hear 'me' more, but in formal writing you will see 'I' more.
I'd suggest that, if you get the question in a test, you choose 'I'.
Student or Learner