I must admit I was surprised when I first read this question, as the words "responsibility" and "care" are two words that I would not have automatically associated with the characters of this excellent novel, particularly Heathcliff and Catherine. In fact, they show every inclination of being extremely irresponsible and careless in the way that they let themselves be governed by emotions and passions.
If you want an example of a character who could be considered to show responsibility and care, I would suggest considering Nelly Dean in the way that she looks after her charges and does her best to care for them, not matter what the circumstances. You might like to consider how she describes her pity of having to give up looking after Hareton when she is sent to Thrushcross Grange and has to leave Wuthering Heights. Although Nelly Dean is a rather interesting character in many ways, we cannot fault her in the way that she loves her charges and tries to look after them as best as she can. She is therefore one of the few characters in this novel that shows responsibility and care.