Both are common. There's a subtle difference in tone. "I broke my arm" is in the active voice and suggests that the speaker takes responsibility for the event. Who or what broke my arm? I broke my arm. "My arm was broken" is in the passive voice, and avoids assigning a cause or a blame. My arm was broken. Who or what broke it? We don't know. It just was. It got broken, somehow.

As Dan says "broken" as in My arm was broken" can be analysed as a passive with no agent identified. But "broken" can also be considered an adjective, and in that case, it describes the state of the arm rather than what happened to it. For example, you might say "My arm was broken, so I couldn't lift my luggage", which indicates that this was all in the past. It's possible the arm is still broken now, or it may have healed, but the sentence doesn't consider that as being relevant to the point being made.

I broke my arm when it's still broken and My arm was broken when it's healed