What is the correct way to texture a cave for a game engine It is not a good thing to make a high quality cave walls for a game engine.
If I create in  Blender a wall for any room, I can create high poly model and then bake a texture for it, then I can use it for low-poly version of this wall.
But it is possible for "small" objects. It could be one wall of big room, but what should I do with long caverns? I cannot UV them and cannot use Blender modifiers in Unity engine to make distortion of the walls and so on...
So I ask anybody, who can give me an advise - what is the best way to create cave texture?
Depending on what kind of game engine you are using 

big objects like a cave are consider terrain as i know 

so your best solution here is to sculpt and material paint the cave in the game engine

but you can split the cave into smaller objects and work with them

I have seen this used ..even if it give bad lighting results