Since the game offers keybinding menus WITHOUT movement bindings (WHY?!), here is something you can do about it.

Open the file input.ini in
%LOCALAPPDATA%\Dust\Saved\Config\WindowsNoEditor\
using a text editor. Don't forget to make a backup of the original .ini first!

Then add the following lines at the bottom of the file:

AxisMappings=(AxisName="MoveForward",Scale=1.000000,Key=Up)
AxisMappings=(AxisName="MoveBackward",Scale=1.000000,Key=Down)
AxisMappings=(AxisName="MoveLeft",Scale=1.000000,Key=Left)
AxisMappings=(AxisName="MoveRight",Scale=1.000000,Key=Right)
AxisMappings=(AxisName="CameraPitchDown",Scale=1.000000,Key=Gamepad_RightY)
AxisMappings=(AxisName="CameraYawRight",Scale=1.000000,Key=Gamepad_RightX)
AxisMappings=(AxisName="MouseCameraYaw",Scale=1.000000,Key=MouseX)
AxisMappings=(AxisName="MouseCameraPitch",Scale=1.000000,Key=MouseY)
AxisMappings=(AxisName="MoveBackward",Scale=1.000000,Key=S)
AxisMappings=(AxisName="MoveLeft",Scale=1.000000,Key=A)
AxisMappings=(AxisName="MoveBackward",Scale=1.000000,Key=Gamepad_LeftY)
AxisMappings=(AxisName="MoveLeft",Scale=1.000000,Key=Gamepad_LeftX)
AxisMappings=(AxisName="CameraPitchUp",Scale=1.000000,Key=Gamepad_RightY)
AxisMappings=(AxisName="CameraYawLeft",Scale=1.000000,Key=Gamepad_RightX)
AxisMappings=(AxisName="PhotoMode_MoveForward",Scale=1.000000,Key=W)
AxisMappings=(AxisName="PhotoMode_MoveForward",Scale=1.000000,Key=Gamepad_LeftY)
AxisMappings=(AxisName="PhotoMode_MoveBackward",Scale=1.000000,Key=S)
AxisMappings=(AxisName="PhotoMode_MoveLeft",Scale=1.000000,Key=A)
AxisMappings=(AxisName="PhotoMode_MoveRight",Scale=1.000000,Key=D)
AxisMappings=(AxisName="PhotoMode_CameraYawRight",Scale=1.000000,Key=Gamepad_RightX)
AxisMappings=(AxisName="PhotoMode_CameraYawLeft",Scale=-1.000000,Key=Gamepad_RightX)
AxisMappings=(AxisName="PhotoMode_CameraPitchUp",Scale=-1.000000,Key=Gamepad_RightY)
AxisMappings=(AxisName="PhotoMode_CameraPitchDown",Scale=1.000000,Key=Gamepad_RightY)
AxisMappings=(AxisName="PhotoMode_MoveBackward",Scale=1.000000,Key=Gamepad_LeftY)
AxisMappings=(AxisName="PhotoMode_MoveLeft",Scale=1.000000,Key=Gamepad_LeftX)
AxisMappings=(AxisName="PhotoMode_MoveRight",Scale=1.000000,Key=Gamepad_RightX)
AxisMappings=(AxisName="MenuXAxis",Scale=1.000000,Key=Gamepad_LeftX)
AxisMappings=(AxisName="MenuYAxis",Scale=1.000000,Key=Gamepad_LeftY)
AxisMappings=(AxisName="MenuDpadAxis",Scale=1.000000,Key=GenericUSBController_Axis5)
AxisMappings=(AxisName="PhotoMode_CameraMoveUpZ",Scale=1.000000,Key=T)
AxisMappings=(AxisName="PhotoMode_CameraMoveDownZ",Scale=1.000000,Key=G)
AxisMappings=(AxisName="PhotoMode_CameraMoveDownZ",Scale=1.000000,Key=Gamepad_LeftTriggerAxis)
AxisMappings=(AxisName="PhotoMode_CameraMoveUpZ",Scale=1.000000,Key=Gamepad_RightTriggerAxis)

These are the settings for using the arrow keys but you can put your own keys in there!

All credits go to JediKnight from the Steam forums! :)