I used to bank in Phuket Town but now with the road works access is a total disaster. My personal solution is that I drive up to the big new shopping mall at Thalang. It is on the right as you are heading north. There are traffic lights so that you can do a u turn easy, and there is a huge car park where you can always find a space. Inside there are branches for all of the Thai banks. The food court there is also pretty good for a cheap and cheery lunch 😆