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.
Plants and gardens have always been a big part of my life. I can remember helping my Dad to prick out seedlings, even before I could see over the top of the potting bench. As an adult, I trained at Writtle College where I received my degree, BSc. (Hons) Horticulture. After working in a specialist plantsman's nursery, and later, as a consulting arboriculturalist, I joined Thompson & Morgan in 2008. Initially looking after the grounds and coordinating the plant trials, I now support the web team offering horticultural advice online.