Should a comma be used before Jr and Sr in a person's name?

Yes. Example: "Martin Luther King, Jr."

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If the name is written in the usual order (e.g., "Martin Luther King Jr."), you may or may not use the comma, depending on your personal preference. But if you're writing for a publication, it will depend on the house style of that publisher.

If the name is written in the reverse order (e.g., "King, Martin Luther, Jr."), in alphabetical lists, for example, use the comma.

Although Jr., Sr., etc., are sometimes regarded as parenthetic, they are part of the legal name and should not be set off with commas.

Examples:

John Wilson Jr.

Bob Jones Sr.

James Stowers III