The Bluetooth module can be a little confusing because there are so many different versions of it (as well as because most of the documentation is poorly translated from Chinese), so there's a lot of conflicting information out there!
The RX&TX pins on the Bluetooth IC on the board have their max voltage specified as 3.3V, and 1.06 version of the board you purchased does not have a 5V-3.3V level shifter on board. So, technically, you should use a level shifter. Many people, either out of ignorance or because they like rolling the dice, have interfaced this version of the board directly with 5V and to my knowledge haven't experienced any problems, but it's possible at some point it could cause damage.
So, yes, with this version of the board, if you want to be 100% safe you should use a level shifter. A voltage divider as you have shown will probably work fine as long as you are operating at fairly slow serial speeds. If you find the signal is being distorted (such as if you have missing or wrong characters), try using lower resistor values such as 1k and 2k, which will allow the signal to change more quickly (using a resistor voltage divider introduces an RC filter, which will cause a little distortion of the signal).
There is also a version of the Bluetooth board that has a level shifter built-in. I actually just got some in stock and will be replacing the current version I'm selling with them soon.
I've been planning on writing a tutorial on using the Bluetooth module, so if you need any help getting it working let me know and I'll push it to the top of my to-do list!