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.
I would also suggest that you apply a treatment of Bordeaux Mixture as a preventative measure. This is a protective fungicide which helps to prevent blight. Blight is one of the most common fungal tomato problems and occurs particularly during warm wet conditions. You can find this product on our website: Bordeaux Mixture.
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