The law setting up SSSIs is the Wildlife and Countryside Act 1981 (c. 69) (search for it at
http://www.statutelaw.gov.uk/Home.aspx ). It states "...of the opinion that any area of land is of special interest by reason of any of its flora, fauna, or geological or physiographical features...". So the first key feature is the specified area of land.
Now as I understand it, land is defined as being not just the surface but also all the way down to the centre of the earth and up to the boundary with space (though I can't cite a basis for this statement). [And I am going to ignore the possible controversy about mineral & light rights etc.] So by specifying the land, one might think one captures the cave under it. More importantly if the bit of the cave is outside the specified land, then it is not caught.
However the specification needs to include the feature of interest, so unless the cave is named (and I would suggest identified by say a survey otherwise what do you do about two caves which are linked but with different names) the cave is not necessarily caught within the SSSI. But the other feature is that the SSSI only has power to stop operations which are specified (they used to be called the Potentially Damaging Operations). So unless the operation is specified (say like not disturbing sediment layers), there is no safeguard for the cave or land from that operation. I won't get into the wonders of how one can side step such defined operations, save to say that SSSI status does not mean the cave is safe even from operations on the SSSI (what about filling in the entrance?), let alone operations off the SSSI which impact on it (like catchment areas).
If I recall correctly, the problem with the Mendip designation was due to the proposed list of operations being much much larger than anyone ever imagined and it was claimed that the list would have stopped the farmers using the land for agricultural purposes.
I should add that there is also scheduled ancient monuments law which can help / hinder (the Assynt Bone Caves has this added complication) and that the law is slightly different between England and Wales and Scotland. Lastly, I am not a lawyer so don't rely on this advice in a legal battle.