낡은 and 오래된 are used to describe objects being old, but never people. If you intend to express, use 나이가 많다, 연로하다, 늙다, 나이가 들다.
- 나이가 들다.
It means a person starting to be old(entering the old age).
Ex: 그 분은 나이가 들고 있습니다.
It tends to be used to mean a stage in life, and rarely used to describe an old individual.
For example: 인간은 왜 늙고 죽을까요? -> Why would people become old and die?
- 나이가 많다
This one is more respectful, especially when you use this to address themselves.
Ex: 여러분 나이가 많은 걸 제가 잘 알고 있다.
This one tend to be used in official writings, and used to describe those old people in general. This is the most respectful, but rarely used in speech or using this word to describe the listener while listener being old.
Ex(in article): 연로한 자에게 관심을 보였던 적이 있으세요?
-> Have you tried caring the elderly?
I am not familiar with them. Please correct my mistakes if there are.