To activate a magic item (DMG p.141):
Activating some magic items requires a user to do
something special, such as holding the item and uttering
a command word. The description of each item category
or individual item details how an item is activated.
Slippers of Spider Climbing are a wondrous item that must be worn on the feet and attuned to be used. Specifically:
While you wear these light shoes, you can move up, down and across vertical surfaces and upside down along ceilings, while leaving your hands free.
Moving is how the slippers are activated.
Now, from PHB p.192:
- An unconscious creature is incapacitated (see the condition), can’t move or speak, and is unaware of its surroundings
- The creature drops whatever it’s holding and falls prone.
So, there are two things to consider - if you have moved and climbed up the wall, you have activated the slippers. If you are unconscious you are unable to move, I think this is subtly different from being able to and choosing not to. Do they keep operating from round to round if you cannot move? Well that is a matter for each group to decide as the rules are unclear.
Notwithstanding, even if you decide that they do keep operating, an unconscious creature "falls prone". In the section on Being Prone (PHB p.190):
Combatants often find themselves lying on the ground, either because they are knocked down or because they throw themselves down. In the game, they are prone,
a condition described in appendix A.
Prone = Lying down
A person lying down does not generally have the soles of their feet on the surface. This is not definitive, however, it is suggestive. Again, each group must make their own decision.