It sounds as though they may be suffering from a fungal disease. If it is just the stems (and not the leaves) that are going mouldy then the cause may be a stem canker. This tends to be the result of fungal spores entering a wound in the stem (possibly where the side shoots removed).
Fungal problems tend to be exacerbated by humid conditions. From your photo it looks like they are in a greenhouse. A good preventative measure would be to increase the ventilation by leaving the door and windows open and watering in the evening rather than the morning to reduce humidity.