How are you getting to Phuket? Seems like the majority of the issues are in Bangkok proper, and if you are flying into Bangkok, there is a potential for some issue. The international airport (Suvarnabhumi) is pretty divorced from downtown (it's a solid 1 hour drive) but if you flying NokAir from Bangkok to Phuket, IIRC it flies out of the old airport which is much closer to downtown...which could be problematic. Basically, stay as far from downtown Bangkok as you can....or hopefully you are going directly to Phuket. But if you are landing in Bangkok, traffic is terrible even in the best of times, so make sure you have
plenty of time to get to your destination and thru security. Personally, I would avoid public transport from Bangkok to Phuket, but that's me.
I would make sure you have trip insurance, DAN insurance and a backup plan...you never know.
Once you are in Phuket and on the water, you are probably the most safe you could be!
There has been political (separatist) unrest in the southern 2 regions (e.g. Yala province) for the past couple of years...I would absolutely avoid going there as you never know if they could use the protests in Bangkok as a reason to step up their violence. You won't be going there as part of your trip, but in case you planned on doing any exploring before and after.
This may help....
What Bangkok State Of Emergency Means For Thailand Tourists
This kind of thing happens all the time in Africa. You just have to be very situationally aware. Stay clear of the hotspots, try to blend in (e.g. avoid really touristy stuff), avoid and/or be very vigiliant during public transport and try to get plugged into the expat community.
Good luck!