In the words of the Hitchhiker's Guide to the Galaxy, "Don't Panic!"
You have to assume fungus spores are everywhere, for the most part, and that they are only looking for an opportunity to grow. That opportunity is usually increased by high humidity, rain - particularly rain that sits on the leaves for days, shade/lack of sunlight, and species. If you irrigate, it can also be triggered by over-irrigation, irrigating your foliage too much, or aspects of your water chemistry.
The easiest thing to do is relocate your bonsai benches so they don't stay so wet. Don't place them under tree canopies - particularly if you know those landscape trees have fungus. When you water, try to water the soil only, and do not water the foliage. Make sure you aren't stacking your trees too closely so that they don't get air/wind. Yes, you can treat chemically - and usually it is enough to treat in the spring when the foliage is young and tender, because once the foliage hardens it is much more resistant to fungus. Systemic fungicides are easiest, but you can also use a rotation of several topical fungicides that will tend to alleviate problems for the season (I use a treatment of Heritage, Clearys 3336 and Mancozeb - 7 days apart - once each spring as buds were swelling).
Additionally, consider species that are more fungus resistant. Conifers tend to be (depending on the fungus) and deciduous trees that have thicker leaf cuticles will be. Chinese elms are so strong that even if they get fungus, they tend to outgrow it. Ficus, silverberries, olives, live oaks, azaleas, camellias, gardenias, hollies, black gums... the list is quite long.
Also note - many trees that get fungus in the spring will not get it in the summer once temps warm, the sun comes out, and rain becomes less frequent. Trees like Japanese maples can be defoliated, sprayed, and the new push of foliage will be fungus-free.
Fungus tends to be an annoyance, and doesn't often kill a tree, though it looks bad. Treat the cause of the fungus, and not the symptoms.
…using species that do well where you live. DAS are not good, JBP are not easy beginner trees, but elms, maples and Shimpaku junipers are typically forgiving.
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