For starters, kitchens (and baths) are the humid parts of any house. That's where all the cooking happens. That's why it's important to have a hood vent in your kitchen. If you don't, that might be a big part of the problem right there...you're just dumping a lot of humidity into the air when you cook.
As you don't have any heating in the kitchen, that likely exasperates things as there's no air flow via the HVAC. So what humidity is in the kitchen stays in the kitchen.
Finally, if the walls are cold, that means there likely isn't adequate insulation in them. So what's happening is your humid air hits the cold wall in your cupboards, condenses, and then keeps them permanently wet, allowing mold to have a good ol' time.
As for how to fix this, the proper solution will be some combination of
- better insulation
- better ventilation
- better air flow
Alas, that's not a small task.
Short term solution may to simply keep your window open in the kitchen when cooking. It will, of course, affect your heating bill. Also consider removing your cupboard doors. That will at least help with air flow so moisture doesn't get trapped in the cupboard.