What major should gene editor study? Do I have to have a master's degree or a doctor's degree to participate in the research?

When you say gene editing, you mean the experimental operation related to molecular biology.

If you major in molecular biology, in fact, other majors will also be applicable to such experimental operations, microbiology and so on.

Some undergraduates also do some molecular experiments.

The master can do it, and the doctor can do it further.