Why are water waves transverse or up and down? 
why are water waves transverse? why aren't the Longitudinal? 
 A: Water waves have a transverse and a longitudinal component. As I understand it the components are the direction of motion of the wave itself and the motion of individual water molecules. The latter comprises the longitudinal component, because water molecules have circular orbits in the wave. I think this becomes clearer with this image
A: Water waves can be quite complicated, they can be both transverse and longitudinal depending on what type of wave they are. I will list three types of waves that are common:
Capillary waves 
Cappilary waves happen at small amplitudes and wavelengths and they are most likely what's happening in your picture. The restoring force is surface tension, so every part of the wave that's curvy has a force that is trying to make the surface less curvy. This creates waves that are mainly transverse.
Surface waves
Surface wave is a name that usually decribes a typical ocean wave. These happen when the wavelength is smaller than the depth of the ocean so they also fall under the category 'deep waves'. These deep waves show complicated behaviour: they are not perfect sinusoidals but are more peaked and the motion is a combination of longitudinal and transverse, resulting in a kind of circular motion as seen in Tatjana's answer.
Shallow water waves
Water waves are shallow when the wavelength is much larger than the depth of the water. Shallow waves are mostly longitudinal. Surprisingly tsunami's are shallow water waves even though the ocean is really deep. The wavelength of a tsunami can reach over 100km. The speed of a shallow wave goes as $v=\sqrt{gd}$ with $g$ the gravitational acceleration and $d$ the depth of the ocean floor. Where the ocean is over 6km deep tsunami's can travel at the speed of commercial airlines. 
A: The surface waves are mostly trasversal because you push water in a localized manner and watch a side effect; the volume waves are mostly longitudinal (sound waves in a water volume below a membrane).
A: Physics is an empirical science, that is an observational science. Waves in matter in general will have both a transverse and longitudinal component.  
Merely by looking at the picture you have posted we can see that the transverse component of motion is far larger than the longitudinal motion on the surface of the water. By continuity, we would generally posit the same in the main body of the water. 
(A more quantitative observation will look at how both these components dampen with time, with distance from the disturbance both along the surface and also in the body of the water. And a theoretical study will try to connect this with the theory of shallow waves). 
