The cousins had no effect on Miss Emily.  The lack of effectiveness of their visit only reinforces the perception of Miss Emily's character.  Her proud, impervious nature forged in a large part by her father is not altered even in the presence of other Griersons.  Emily continues to act as she wishes, and the cousins eventually leave.

The cousins did nothing to improve her mood or help her reconnect with the world outside her home.  The narrator describes the cousins as "even more Grierson than Miss Emily had ever been," meaning that they were even harder edged than she, less friendly and less likely to attract a man.  No one in town likes the cousins, any more than they like Miss Emily.  However, the opinion of the town that she wanted to be rid of them too ("we were all Miss Emily's allies to help circumvent the cousins")  demonstrates how presumptuous the town is towards her feelings.

Still, obviously they did not have a positive effect, as she dies alone, in a hideous way that shows the depth of her loneliness.