I definitely see the Nurse as a likable character. Aside from the humor she adds to the play, her genuine love for Juliet is impressive and heartwarming. She counsels Juliet from the beginning of the play, and I've always found it to be Juliet's exclusion of the Nurse from her plans (she becomes upset when the Nurse tells her she should marry Paris after Romeo is banished) to be a very sad part of the play.
I believe that the Nurse is much like Mercutio in adding comic relief to the play. We enjoy her honest,wholesome, and humorous character against the dark tone of the unlucky lovers. She also acts as an accomplice in the play along with Friar Lawrence. She aids in the development of the events between the two lovers by acting as a messenger. Although Juliet is upset with the Nurse when the Nurse tells her that she may as well marry Paris because Romeo has been banished, the Nurse is true to her practical and honest character in this scene.
The Nurse provides a great deal of the slap-sick comedic relief throughout the play. She is an immensely likable comedic character for the crudeness of some of her comments, her sometimes absent-mindedness, and her devotion to Juliet.
To be sure, the groundlings would have loved the Nurse and Shakespeare likely crafted a good deal of her character so that they would indeed like her. For, in many ways, she was likely much like them.
During the time of the setting of "Romeo and Juliet," it was not uncommon for families to take in their indigent relatives to act as nurses, etc. for the children. It is highly likely that the Nurse of Shakespeare's play is such a character. And, much like the doting spinster aunt, the Nurse is effusively affectionate:
Thou wast the prettiest bab that e'er I nursed./An I might live to see thee married once,/I have my wish. (I,iv,35-37)
When she tells Romeo at the party the identity of Juliet, she reiterates her feelings, even boasting:
I nursed her daughter, that you talked withal/I tell you, he that can lay hold of her/Shall have the chinks. [cash](I,v,111-112)
Clearly, the Nurse, with her bawdy language and her bumbling mannerisms is character that provides comic relief. In Act II, Scene 4 as she enters with her own servant, Peter, who carries her superfluity of clothing, Mercutio shouts, "A sail, a sail!" And, when she calls for her fan, Mercutio says,
good Peter, to hide her face, but her fan's the fairer face. (II,iv,33)
More comedy is provided as the Nurse purposely delays telling Juliet what Romeo has said after she is sent upon an errand in Act II, Scene 6;
Jesu, what haste? can you not stay a while?/Do you not see that I am out of breath?
To this Juliet replies,
How art thou out of beath when thou hast breath/To say to me that thou art out of breath? (II, v,31-32)
I think it would be hard not to like the Nurse. I think this is in part due to the fact that she is written as a comical character.
Her wit is part of why I like her, as is her somewhat bumbling nature. For example, the first time we see her, she keeps talking and talking so much that Lady Capulet keeps having to tell her to shut up.
Beyond being funny, though, she really seems to care for Juliet. She is kind to Juliet, even more so than Juliet's parents. She stands up to Lord Capulet when he gets mad at Juliet for not wanting to marry Paris. She tries to be sure Romeo is good enough for Juliet.
So she is a funny character who also truly cares about Juliet -- so what's not to like?