Which word is more idiomatic with a problem, to do or to make?
I don't like to do/make problems for people.
To do problems isn't idiomatic, and when you use it, it carries a meaning of solving the (homework) problems. I don't think that's what you intended.
Make is better, but even more idiomatic are cause and create. This is supported by:
the Online OXFORD Collocation Dictionary:
bring, cause, create Success brings its own problems. Staff shortages cause problems for the organization.