The water is coming from somewhere. Is it a leak? A damp floor? Condensation from an adjacent steamy room? Wet laundry?
I suspect the op is going to paint the cupboard whatever anyone says.
You can use a Dulux Damp Seal. This is available as water-based, which will be more pleasant to apply in such a confined area - and allow to get the emulsion on quicker.
No doubt oil-based/solvent based versions are still available.
Anyhow, this will not solve your problem.
It will only mask the problem.
You might get a year out of it. You might not get five minutes out of it.
Anyway, back to JohnD.
Good. Glad I'm wrong.So it's under my neighbour's stairs, the stairs which go up to his flat. Perhaps because there's no ventilation in the cupboard itself that's why it is getting damp? Would drilling a few holes in the outside wall help?
The OP is reading and taking advice, so you suspect wrongly.
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