For a damp basement, it's better to combine insulation with waterproofing - neither one alone will do the trick. Start with sealing: if your basement is leaking, deal with moisture first - seal cracks in the floor and walls, treat them with penetrating waterproofing like liquid glass or bitumen mastic. Then add a waterproofing layer (roofing felt, polyethylene film - something that won't let water through the walls). Only after that can you insulate.
As for materials: I wouldn't recommend mineral wool in a damp basement - it absorbs moisture and loses its properties. Foam or extruded polystyrene - now that's what works! Extruded polystyrene is actually waterproof, doesn't rot, and keeps its insulating properties for years. Attach it right to the waterproofing layer with adhesive or dowels, then cover it with a vapor barrier film so moisture from the room side doesn't get into the insulation.
The thickness of insulation depends on how cold your winters get, but on average 50 - 100 mm should be enough for a vegetable storage. The key thing is don't break the layer - seal all the seams with foam. And don't forget about ventilation in the basement, or condensation will start building up and all your work will be for nothing! A few small vents or a supply-and-exhaust system - and your vegetables will store great.