Is this Moore Diagram designed to identify a sequence Correct?
I've picked up on just one mistake. From E7, when input is "B", it should go to E2, not E4. E4 is supposed to have detected the sequence "BAB", but the way you have connected this would relate to a "BAAB" sequence. All else seems fine.

state | sequence | in = A | in = B
==================================
  E1  | ""       |   E1   |   E2
  E2  | "B"      |   E3   |   E2
  E3  | "BA"     |   E1   |   E4
  E4  | "BAB"    |   E3   |   E5
  E5  | "BABB"   |   E6   |   E2
  E6  | "BABBA"  |   E7   |   E4
  E7  | "BABBAA" |   E1   |   E2