Stupid question, but were you sure it was mold and not just tiny root hairs? https://freshsprouts.net/mouldy-sprouts-and-fungus-on-microgreens/
I’ve grown quite some batches of broccoli and never had any problems (that I’m aware of). Though I do keep my jar in a dark cupboard instead of the sink. I just let it drain in a bowl. The jar should be tilted, but not completely upside down