It is understandable that you have concerns regarding safety in Kenya, especially after the terrorist attacks that took place last year. There are certain precautions you will need to take when moving to Mombasa. For instance you will find that most expats hire security guards to protect their homes as crime is high especially in the cities.
To avoid any political unrest, you should steer clear of any political rallys or demonstrations.
For a more detailed insight check out the Safety in Kenya page.