Botrytis is a big problem here, but I admit to not doing much about it, because living with it means that my seedlings - the ones that thrive - are that much more resistant to it. But I have been known to break and apply a copper spray early in the season if I get really worried that really little seedling bulbs will be wiped out before they get a chance to put on any weight. I think it slows them down, so that many don't flower until their third or fourth summers rather than second, which is a torment when I'm just busting to see the reslt of a cross! Then I remind myself when I finally see them, that they're survivors.
But burning old stems is good, if you can do it. Collecting and burning flowers as they fade helps too - less material to harbour the fungus. Lorn and Rick have good advice.