He was greatly concerned that finding out more of her past would hurt Karen and the people close to her. After Risako Nagisa was beaten in episode 2, Karen spoke to him through the hidden listening device that was in Risako's swimsuit and told him she would be taking control of her life before she threw it into the ocean.
After Karen and her friends returned from their voyage, he continued to keep an eye on her from afar and was happy to see her in good spirits.
The Mysterious Man's relationship to the project that cloned Karen or Karen herself is unknown. He was certainly very close to the original Karen that was used as the base for the clones. While he may be her father, it is also possible he is the boy in the photo of child Karen. This would make him either her brother or a very close childhood friend.
Whatever the relationship, he took Karen away from the project and had her memory suppressed in order to help her live a normal life, but was unable to do the same for the other clones. Given his intent to keep Karen away from her past, it is unlikely he is responsible for the butterfly tattoo that lead her back to her origins.