Emily is not crazy, in fact she could be considered eccentric for her lifestyle, but not insane. She did commit murder, however, so she is unstable. In her desperation, she decides that Homer Barron is her last chance for companionship. She knows that he prefers men, so she just wants someone in the house with her. Sad, really, that she resorts to murder.
However, Miss Emily is a symbol the old South that refused to modernize and realize that the world was created anew when they lost the war. She is a sympathetic figure, not crazy.
"We believed she had to do that" is the quote which supports that Emily isn't crazy or insane. She simply lives in a world where her reputation is the end-all be-all. We know from the many suitors she had that she wished to be married. However, her father turned them all away as unsuitable suitors for his little girl. Only after her father's death was she able to find her love, and when he indicated that she was not suitable for him (speculation, of course) she kept him where he needed to be in order to retain her respectability in the community. In her world, she is perfectly sane.
From our point of view, perhaps she is insane. Dealing with disappointments in life is part of everyone's life. We have all been taught that killing is wrong, and I'm sure she also had that lesson. However, her desire for companionship and to keep her reputation in tact outweighed her fear of the consequences.
For someone to actually commit a murder and one that is premeditated is a clear indication that he/she is mentally ill, obviously, but because Emily had such a troublesome and controlling upbringing, the reader can sympathize with her plight (although some readers do not, despite the narrator's attempts to pay tribute to her and not pass judgment on her).
I am not blind to the fact that Emily committed an unspeakable crime that was horrific and calculated. She was clearly wrong to do so and had she not died without it being found out, she deserved harsh punishment for it. However, I sympathize with her character because her father was so very controlling and never allowed her to be her own person or to grow up normally, experiencing what most children and teenagers would.