How do you justify to your staff the fact that Muffy is making a higher salary than any of them and that they did not have the chance to interview for the position?

There are two answers that immediately come to mind - I can't or she was way qualified... From the perspective of I can't, I would focus on two specific points. Either she is getting too much or the staff is getting too little. The results of this hire already created morale issues that undermine relations. It will be critical to make sure that the staff understands that they are being treated in a fair and consistent manner and that the lower-paid employees are getting paid at the appropriate level that incorporates a full range of upper and lower limits for their particular jobs.

The second point would be to demonstrate her credentials including relevant experience, knowledge and potential to increase the department from previous jobs and assignments output. This would go a long way in showing why she was hired over the existing employees and that it had little to nothing to do with her spouse being the chief. Again, a sound job description and the details of hiring from either the civil process or some other documented hiring procedure and policy could justify her coming on board.

How does the personnel office handle this problem in light of the city's civil service system?

As noted, compliance issues with regard to hiring should have been taken into consideration....

For example, the ADA would prohibit discrimination against all qualified applicants with disabilities. Determining the essential functions of the job before beginning the interview process would have been a more professional approach. If any of the staff were disabled but felt that he or she could have don the database administrator job for instance, that individual may have had a viable legal position was given to the spouse of the chief even though he was qualified and therefore discriminated against. But it is not just the disabled because the Chief's wife got the job seemingly with no other outside competition.
In order to stop the toxicity of the staff - the human resource function could establish a mentoring program that would force the existing members to be responsible for the training and integration of the new database administrator. It would also be appropriate to carefully screen the entire new hire process to ensure no laws or policies were broken as well as reviewing the range of the new salary to ensure it fell in the proper range for the position and experience. This in turn would guarantee that new administrator was compensated appropriately. Like compensation, verifying that the new hire had the appropriate technical skills as well as a thorough discovery on educational credentials and previous employers. All in all, the human resource function would need to verify if this was in fact a good and legal hire.