Assuming that the caravan is watertight and has not previously suffered water ingress into the fabric of the walls or floor, the smell of damp will most likely be coming from the soft furnishings: seat cushions, carpets, curtains or mattresses.
Step 1: Machine wash any upholstery that can be taken out of your camper. Choose a cool and gentle wash to prevent damage to old or sun damaged fabric. Also use Dettol Laundry Cleanser instead of fabric conditioner. Line dry in strong sunshine and finally steam iron everything you can to kill any lingering mould or mildew spores.
Old caravans/motorhomes often develop a musty smell because of poor design of the seat cushions and bases, which rarely have enough ventilation. Whilst you sleep moisture travels through the mattress and condenses on the base of the bed or slats, which enables mildew and mould growth.
Due to space and budget constraints most new leisure vehicles still do not have adequate seat base ventilation. For example, my 2017 motorhome suffered terrible condensation build-up under the bed until I fitted BedAirer™ mattress underlay.
Step 2: If you are able to remove the caravan seat covers and launder them, replace the foam for new. The old foam will have lost its spring years ago. To achieve the best results, sandwich the new foam between two layers of BedAirer™, the springy layers will improve comfort and significantly reduce moisture build up in the seat cushions, aiding better sleep. The bottom layer of BedAirer™ may be sufficient to prevent condensation building up on the seat bases. If not, or if you were unable to add the breather layers to the seat cushions, then add an underlay layer of BedAirer™ which will allow free movement of air between the seat and the cushions.
Step 3: Remove and replace any carpets. I recommend replacing them with removable washable door mats such as Turtle Mats or Marvel mats from Dunelm Mill. I have used Marvel mats in three motorhomes and found them to be very easy to cut to size. They do not fray, but do release a lot of carpet fibre initially.
Step 4: Clean any non-removable fabrics. This is a 3 step process: Mix 1 part white vinegar with 2 parts warm water in a spray bottle and dampen the upholstery. Sprinkle bicarbonate of soda on or rub into vertical surfaces. The vinegar and soda will react. Leave for 24 hours to dry then vacuum clean. Ventilate well.
Source: wikiHow where there are detailed instructions about the use of vinegar and bicarbonate of soda on many materials.
Step 5: Clean all hard surfaces with Wilko Mould and Mildew Remover or a similar product. Damp and musty smells can be significantly reduced by washing porous surfaces with a 1 to 1 ratio of white or cider vinegar and water.
Step 6: Ventilate, Ventilate, Ventilate; both when in your motorhome and during layup. During cooking and showering it is essential to have the caravan windows or top vents open, so it may be necessary to run the heating at the same time. During layup remove as much as you can from the caravan and leave all of the locker doors open. Dehumidifiers are great at keeping leisure vehicles dry throughout the winter, but caravans and motorhomes have ventilation built in so dehumidifiers have to work quite hard. Again, vinegar can help reduce any lingering damp musty smells in caravans. Simply leave a bowl of vinegar on the kitchen work surface and it will absorb the mildew smell.