This is a divisive issue, and different communions (or churches) in Christianity would have different approaches to this issue.
Biblical theology teaches that animals are there to serve man, beasts of burden haul the plough, and the drays loaded with produce.
Christian theolgoy has changed its point of view on the animal kingdom in recent years due cloning of animals, gene sharing, and understanding of life.
Pope John Paul II, the great philosopher pope and world statesman, taught that animals have souls and have to be treated with the respect that is properly due to them.
Mother Teresa said that all animals give glory to Christ, and reflect the spirit of God. Animals should be treated as one would treat Christ in the poorest of the poor.
Fox hunting is a sport which is coming in for much community resistance and vilification. It is true that foxes may be declared verimin, and yes, they hunt and kill other animals. However, animals do not have consciousness like humans, and do not have the faculty of reason, so must follow their own instinct.
Where there is danger to human life, it is OK to kill an animal. Where there is danger to livestock and property, it is OK to kill an animal. Think of farmers shooting feral cats and wild dogs which attack and kill sheep. These things are necessary, needful and follow moral reason.
Hunting is another matter, and probably, will cease to be a sport when sufficient community pressure about this is exerted in the media.