Any source of comfort and calm can be a healing agent for someone who needs it. For many, an animal friend will be their closest companion and petting and cuddling them will contribute to their sense of well being. This can bring on more health or even provide some healing in a time of illness or at least be palliative care.
Yes, animals have an influence on human health, because they can offer comfort and healing. Hospitals know this, and offer healing dogs that make rounds to visit sick people. Animals can offer great companionship to their owners. Animals can be a negative influence, too, if they are sick and bring diseases or are even unclean and not cared for.