We are never actually told how it is that the family, and the mother in particular, knows that the letter contains bad news. What we are told is that the mother has a strange premonition that the letter will tell her something terrible. Perhaps this premonition can be explained by the intense love that she has for her son, the dead Pete, and the unfathomable bond that exists between mother and child. Note what the poem tells us about her instinct and intuition:
Fast as she can she hurries, something ominous,
her steps trembling,
She does not tarry to smooth her hair nor
adjust her cap.
There is no indication, from what we are told in the poem, that would lead us to suspect that the letter contains bad news. There is just the mother's instinctive reaction to having received this letter that points towards some kind of deeper intuition or premonition.