Sorry, I don't have information internally as to the DJM-707's multi-voltage status, so to play it safe, I'll recommend you only use whatever input voltage is listed on the back of the mixer. In your case, that should be 240V. To use it in the US, you are correct - you'd need a step-up transformer to go from 120V source to 240V device.
Looking at the transformer, it seems you've got it set correctly; the switch is to select the source voltage (120V), and it would automatically set itself as a step-up transformer.
It can handle 300W, which the DJM-707 is far below, so it it's not turning on, it could certainly be a fuse. Having opened a 707 before, I can't recall if there is a fuse inside, but I wouldn't recommend you open it yourself - that thing is a crazy sandwich of circuitboards.
You can contact one of our Pioneer DJ authorized service centres to have your mixer serviced.