oversensitive; thin-skinned; high-strung
Describes a person who is easily hurt, worried, or stressed by small things; often used critically or self-deprecatingly. The variant 神経の細い is also common.
See also: 神経の細い
He's oversensitive, so he gets depressed over the slightest thing.
神経が細い人は、この仕事には向いていないかもしれない。
A high-strung person might not be suited for this job.
デリケート is a loanword meaning 'delicate' or 'sensitive', often used for both people and situations, but 神経が細い is more colloquial and carries a slightly negative nuance of being too sensitive.
The expression literally means 'one's nerves are thin', metaphorically suggesting a fragile or easily affected mental state. The exact origin is uncertain, but it is a common idiomatic phrase in modern Japanese.