Is the Scrum Master responsible for starting and stopping Sprints?

That's because the answer is no -- it isn't within a Scrum Master's gift to start or stop a Sprint.

The Scrum Guide says: "They are fixed length events of one month or less to create consistency. A new Sprint starts immediately after the conclusion of the previous Sprint."

No one starts and stops Sprints. They start and stop on a cadence.

However, depending on the tools that the team is using, there may be a need to "start" a Sprint in the tool, doing things like marking the initial state of the Sprint Backlog for measurement and metrics purposes. The Scrum Master may be the one to do things like this, but I would worry that the team would view the Scrum Master as a secretary or tool administrator rather than a coach and facilitator.

As others have said, the Sprint starts and stops on cadence. 



I would add that there is a case that a Sprint would end outside of regular cadence, and that is because the sprint goal is no longer relevant or valuable. 



In that case, the PO is the only person with authority to cancel a sprint. 

Just as everyone has said, Sprints stop and start automatically.  However, if you want to designate specific triggers I would use the following. 

The Scrum Team ends a Sprint when they complete their Sprint Retrospective.  The Scrum Team starts a Sprint when they start Sprint Planning. 

As @Thomas points out, if you have some type of tool you use to manage your Product and Sprint Backlogs, there may be a need for someone to initiate the action to start/end a Sprint.  Who does that is up to your teams to decide but I usually advocate that it shouldn't be the Scrum Master because that gives an impression that they are in control.