I always seem to get black onion aphids in my deep water culture, and flood and drain beds affecting my green onions and garlic chives. They also attacked my terrestrial garlic.
The deep water culture floating raft ones I can blast off with a strong spray of water (but they come back quickly), but I can only wash them into the grow media (hydroton) in the grow bed as pictured and they bounce right back.
Neem and soap are no good for my gold fish. What else can I use that is fish friendly?
This is a provisional and untested answer. Claims with some scientific trials as data do show that the pest load on plants is reduced with the use of foliar sprays of actively aerated compost tea (AACT). Vermicompost tea may be even better. The bacteria and fungi that is applied to the leaf surfaces may be able to kill aphids and is unlikely to harm the fish.
E Coli populations present in AACT are thought to vanish at about 72 hours of incubation due to the competition from the other organisms in the brew.
Since it's being sprayed onto leaf vegetables, you'll want to cook the vegetables before eating to be safe.
As always do your own research before trying this out in your own aquaponics system.
Edit: 19th Jan 2017
https://www.google.com/patents/US6475503
In brief vermicompost contains organisms which release chitinases which dissolve insect skeletons. Spraying leaves with vermicompost tea drives insects away, and it may be incorporated into hydroponic solutions. Also, feeding plants vermicompost may increase the level of plant production of chitinases which again protects plants from insects.