How do floating wind turbines stay upright? Some designs for floating offshore wind turbines have asymmetric bases. 
Why, however, is the turbine sometimes based on the downwind side of the float as seen here and here? 
And, related, what direction is the force on the top of a spinning turbine? 
My expectation would be that the force vector is roughly the same as the wind vector, and so the turbine would be attached to the upwind side of the base. This way, the buoyant force on the base could balance against it. 
 A: I think it depends on how the base provides balance (does it floats or it is planted on the seafloor?). Edit: Ok I realized they float!
If it floats, the resistance of the base to tip outside the water (if downwind, the upwind part will emerge if the turbine tilts backwards) is compared with the resistance of the water to make way for the structure if it sinks a little (if the turbine is upwind, when it tilts it would push down the opposite part of the structure). If the center of mass is on the turbine side, for an equal angular displacement in the upwind scenario you push down a larger volume than you make emerge in the downwind case.
Edit: I leave here the discussion of the seafloor basement for the curious reader.
So let's say the base is planted on the seafloor. In that case the answer should depend on how the base is attached on the seafloor. Let's simplify the most and analyse two possibilities: you stabilize the structure with a long, heavy horizontal segment lying on the seafloor upwind or downwind. Putting it downwind will create the problem of slowly digging a well in the seafloor, due to the constant torque coming from the wind. This well would make the structure tilt. So you put the segment upwind (realistically, you care more about having a long, heavy upwind part of the base).
I think it is for this reason that you put the turbine downwind wrt the base's vertical shaft. In this way you make effectively larger the upwind part of the seafloor base.
In other words in this way the torque of the wind will cause less digging of the seafloor and improve the stability of the structure over the time. 
It is true that you could just build the upwind part of the seafloor hinging larger to begin with, but this sure makes it even better.
------back to the floating case:
Note that it is perfectly possible that the placement is chosen regardless of the stability of the entire structure. It could be beneficial in terms of maintenance to keep the tip of the turbine inside the base's shaft perimeter (so that if they have to fix it they can put some scaffolding on the base and access the pivotal part of the turbine with no problem). This also makes sense because the shaft of the turbine (not of its base) is built slightly tilted towards upwind, so that you would need to reach it from the water. 
So maybe it is mainly for practical reasons.
Also you can move the center of mass of the structure with some weights and make it nearly coincide with the geometric center of the floating base, so that at that point there is no difference.
A: The first question to ask about these floating platforms is what provides the horizontal force to balance the drag from wind and keep the turbine from acting as a sail and blowing away downwind.  I presume that the answer is a tether cable anchored to the sea floor.
This leaves the designers with a choice as to the relative positioning of the turbine and the cable attachment points to the platform. A lot would go into that decision, but I assume that the points would be across the platform from each other to create a stable equilibrium point in the rotational position of the platform in strong winds. Otherwise the platform would be prone to twist around frequently, greatly complicating the problem of keeping the turbine pointing into the wind.
I am sure that the turbines are not locked to point in one particular direction, but this tether cable explanation would mean that most of the time, except in strong currents, the turbine would end up on the downwind side.
