At some point, everyone loses a loved one. It's an inevitable part of life. We all worry about it to a certain extent, but for some characters, the fear of someone close to them dying can take over their life. Sometimes it's rooted in the character not wanting to see their loved one suffer, but it can also be centered on uncertainty about death itself, the character's ability to cope on their own, or how their life will change with their beloved no longer in it.

Being overprotective of a loved one's health
Refusing to talk about the possibility of a loved one's death (even if it's likely to happen)
Being morbidly obsessed with death or funereal trappings, such as Victorian death masks and portraits
Limiting a child's freedom and independence to keep them safe—not allowing them to drive, stay out late, or visit new places, for instance
Being obsessed with germs and sanitation
Forcing family members to take part in health crazes that promote longevity
Holding onto things as they are instead of allowing people and situations to evolve
Having panic attacks when thinking about the death of a loved one
Refusing to make contingency plans for a loved one's death (not buying life insurance or making a will, etc.)
Refusing to accept a loved one's illness or terminal diagnosis
Visiting psychics to gain insight into a loved one's well-being and future
Investing in extensive security measures to keep family members safe

Obsessing about a loved one's death and what it would mean for the character
Wanting to hold on tightly to a child while also knowing they need freedom to grow
Letting anxiety and fear control decision-making