To answer your question at a basic level, if you want to change sprint scope while the sprint is running, the backlog items you change need to be an equivalent size. For instance, if you need to add a 5 point story to the sprint, you could remove a 5 point story from the sprint. Or, you could remove one 2 point story and one 3 point story. The sprint commitment therefore stays the same - you're not doing extra work.
You didn't say if you were doing this or not, but don't remove anything from the sprint if it's marked as in progress, as this will disrupt the team's work.
Achieving the sprint goal should be your aim. If there are additional things that you'd like the team to do if they get through the sprint backlog faster than expected, consider having those as stretch goals. With this method, you don't need to add them directly to the sprint and you don't need to take anything out. If the team completes the sprint backlog early, they could take the highest priority item from the list of stretch goals and bring this into the sprint. It needs to be something that can be completed within the sprint - so be careful with adding large items towards the end of the sprint. If the stretch goals are not completed, the sprint has not failed. Stretch goals are additional work - they are not your sprint goals.
Failing a sprint can be very demoralising for the team, and seems like it could be prevented if it is happening because of items added to the sprint that do not relate to the sprint goal.
If stakeholders ask for more work while the sprint is running, all you need to do is say that the work can be considered for a future sprint.
It would be helpful to know what your role is, as you didn't say. That might be another question though.