Claire is a practical woman who likes to take charge. She left to make a life for herself in New York, while Catherine had to quit school to stay home and care for her father. But, Claire did send money to her father and Catherine so they could pay off the house. It's not as if she abandoned the two of them totally. Claire seems to have taken her family responsibilities seriously. I think Claire is very jealous of the relationship between Catherine and their father. Catherine inherited her father's genius for mathematics, and Claire doesn't fit into this world that her father and Catherine share. Claire felt he belonged in an institution, but Catherine totally disagreed. Along with dad's mathematical genius, Catherine seems to have inherited his madness as well, and Claire now believes it is her responsibility to take care of Catherine. Perhaps this is guilt because Claire left? Claire is also the one who seems convinced that Catherine is insane, but even though there are signs, we don't know for sure if this is the case. Does this make Claire a controlling, cold, and insensitive person? I don't think so. She just doesn't know how to relate to her sister, perhaps because they never developed a relationship with each other. Catherine was her father's confidante, not Claire's. There's no doubt that resentment exists on both sides, but underneath it all, I think Claire cares about her sister and wants to do what she feels is best for her.
Claire is often insensitive, and somewhat calculating, but she does not do it to be malicious. Claire has her family's best interest at heart, but she is not skilled at softening her approach.
Claire lived in a small, studio apartment to work and pay off her family's mortgage. She sacrificed her own creature comforts for the family's happiness.
While Claire and Catherine do not get along, Claire truly believes that Catherine would be better off moving to New York. Catherine believes she just enjoys interfering, but Claire really wants the best for her sister.
Claire also believed that both her father and family would be better off by putting their father in an institution. Claire is not being intentionally cold and unfeeling, she thinks it is the most practical solution.