How can I say "I have translation background" in Chinese?
"背景" in Chinese can also be used to refer to people's background. Therefore it is okay to say "我有翻译背景". "经历" is also fine but it is "experience" in English. If you think your "background" means "experience" exactly, it is fine.

In universities we often say "你有计算机（专业）背景吗？" ("Do you have background in Computer Science?") If you have a degree in Computer Science then you definitely do. However, we probably don't think a degree in Computer Science counts as experience.