You’d think someone with a college education would know whether or not to use an apostrophe, but apparently they don’t teach this anymore.
The professor says use an apostrophe. Why you ask, o seeker of knowledge? Because bachelor has two meanings: An unmarried man and someone (male or female) who has earned a college degree. Since the degree is owned by this graduate, an apostrophe is needed to show possession.
One more thing: Lowercase bachelor unless giving the name of a specific degree. So you can have a bachelor’s degree or a Bachelor of Arts Degree.
Bonus for advanced students seeking additional coursework: The same rules apply for masters degree vs. master’s degree.