I have a huge problem with aphids killing my seedlings, and even toppling some of my fully-grown fans. So, I started out by trying a product that I thought was near-organic (non-toxic and biodegradable) but that turned out to have properties that are not organic. Special thanks to Sooby for explaining how to distinguish differences between organic versus non-hazardous insecticides, and for pointing to readily available products that are fully organic (will be trying them next!).
For aphid control, I first tried Dawn Ultra non-antibacterial 2x strength that I bought at the grocery store. I mixed 1.5 Tablespoons of the 2x-strength dishwashing liquid in a gallon of water and transferred it to a spray bottle. For each seedling, fan, or clump, I tried to use up to 3-4 tablespoons worth of the spray (close to 1/4 cup), which turned out to be jjust enough to cover the base and insides of the leaves where aphids like to hide. I waited a few minutes between a first and second application, to catch any aphids that didn't get sprayed directly but that came out of the leaves soon after.
One application of anything can't control adult flying aphids - they just fly in and start laying eggs afterward. And, spraying once might not eliminate any already-laid eggs, either. So, I planned to follow-up by spraying every week for a month or so, until the egg-to-flying stages might seem to be disrupted. And, I'm working on learning more about the many choices out there for fully organic insecticides, some of which I've learned are available at big box stores and online for comparably affordable prices.
Not technically organic: Dawn Ultra non-antibiotic
Certified Organic: Ortho Elementals and Safer's Insect Soap