When one development team member is not getting along with the other development team members

If one development team member is not getting along with the other development team members, what should the Scrum Master do?

Assume good intent, reinforce mutual professional respect between all parties at all times, and harness any potential for constructive dissent which might be in evidence.

So who is responsible for resolving this?  The development team or the scrum master?

In Scrum, team members are jointly responsible for resolving such things. The Scrum Master's responsibility is to cause the necessary changes in behavior.

As the Scrum Master, do you recognize the Scrum Values (courage, commitment, focus, openness, respect) within the Scrum Team?

Do you see evidence that the Development Team want to be self-organizing? Are the team aware that this is expected of them? Does the organization empower the Development Team to be self-organizing?

Do the team discuss this problem (perhaps at a Sprint Retrospective)? If not, have you tried to find out why?

My pragmatic answer is that an experienced Scrum Master (somebody who has experience dealing with difficult personalities), or somebody with experience as a credentialed professional Coach (likely from Organizational Development or HR - NOT "Agile Coach") would have some one-on-ones with the person in question. There could be underlying issues that is affecting their performance and their work ethic. With time, and nurture, this should be minimized, or eliminated.

The worst case scenario is that after weeks or months or deteriorating behavior, the line manager is asked to intervene. I know of a situation where the development team member was furious over the transparency that Scrum brought (that is, he did very little work) and that he had no control over the situation, that it took him being transferred to another department to raise the morale of the team.

I know of a situation where the development team member was furious over the transparency that Scrum brought (that is, he did very little work) and that he had no control over the situation, that it took him being transferred to another department to raise the morale of the team.

I had that in one of my teams, too. Took me months of one on ones with him. Ultimately he was afraid to lose his position as the "to go to" guy for a certain area of expertise.  

OT: try to create mutual understand amongst the team members. For instance with journey lines, the COIN method of feedback and assess the level of trust. Work from there. Hope this helps.