How to install baseboard at the transition between floors with different height?
This can be a tough problem to work out in a situation like this. Particularly if you are trying to run the same type and size of molding from one floor area through to the other. What I have done in instances where the height changes are small is to simply trim the bottom of the molding so it fit into place over a distance such that the it slopes from one height to the other. If the slope is small and over enough of a distance it will never be noticed by anyone but you (unless you point it out to others :^) ).

Installing the sloped molding piece does impose some interesting work with making the miter cuts on the ends. It may also involve some planing and sanding of the molding surfaces at the joint to blend it around the corner. (BTW you will be a long way ahead with this type of "blend to fit" with real wooden baseboard. MDF or plastic foam type baseboard fails miserably when this is attempted). 

I see that the existing darker colored floor has baseboard without any base shoe molding. You may want to consider the addition of base shoe molding in the new area because it can play an essential role of covering up what you have to do along the bottom of the baseboard to slope it from one height to the other across the transition zone. It is acceptable to bring the base shoe molding up through the transition zone and then just cut it back off at an angle at the end of its run.