I agree that having a degree is not high on the list of qualifications for being an SM and it definitely is not a requirement.
In my opinion, having a vast amount of theatre experience and knowledge makes the best SM's. My tip would be to do everything. do lights, sound, props, box office stuff, house stuff, design, act, direct, build scenery, etc. Doing a little bit of everything will give you the best experience available for you will get to see how an SM interacts with each person and then take your own personal experiences, separate the good from bad, and figure out how to be the best stage manager you can be. That is how I got started.
I had no intention of being a stage manager, especially being very good at it and liking it. I had done everything in the theatre except stage management through high school, college, and doing community theatre. I never graduated college, but i was asked to come help at a community theatre that I had never volunteered at, with a cast I had never worked with, in a city I had never done theatre in. I had to become the stage manager two weeks before tech rehearsals for a cast of 45 because the director and stage manager was removed from the show for being WAY behind schedule. (it was bad)
Anyways, I took all my knowledge of theatre and I remember the good and bad SM's I had worked with and employed their techniques. (having run successful businesses as my day job helped out some here also). I was told by several people that I was the best stage manager that they had ever had and was immediately asked to do the next show, too.
However, be careful that you completely understand what your director expects of a stage manager before accepting the position. I knew I could do it because I had worked with stage managers in all aspects of theatre and have several good friends that are stage managers.
I do recommend you finish your degree if that is your goal. But do not be afraid that it is required if you decide to focus on another major.
Good Luck!