The Thai government has invested a lot of money in keeping Phuket safe and attractive to tourists. This means swift justice for crimes and has resulted in very low crime rates. You will also see a lot of police, around while driving, which they are always standby to enforce the law.
If you are planning on renting transportation like a motorcycle or a car, please be aware that there are some dangerous drivers on the roads and many people drive drunk when coming back from bars.
Of course, like anywhere, keep your wits about you and don’t draw attention to yourself. If you worry about the political problems in Bangkok, be informed that these problems generally do not affect Phuket at all.