It easy to see why these can be such a problem. With over 4,000 total species there are a whopping 350 or so breeds that are damaging to agriculture and forestry. Even with a fully indoor system these insects can ride in on a shirt, via the breeze, or even on plants themselves if you purchase already developed plants. The requirements for them to breed is relatively limited because they, or at least the majority of them, are asexual. This means that they can reproduce with a single source thus making it easy for their populations to grow.
Rather than read about it - here's a video. Check out the article too!
Nate Storey - Algae in Aquaponics
Given the temperature and moisture required to effectively grow plants, it creates a perfect breeding ground for things like mildew. Mildew is quite widespread in the indoor Ag industry for these exact reasons - because of high density farms with limited circulation. It can be spotted with the eye and looks like a white or grey powder spot(s) on your plants. There are several ways to help prevent mildew from forming, including; finding disease resistant strains, providing adequate circulation, minimal shade on plants and minimizing fertilizer (use slow-releasing).
Check out Home Style Aquaponics for a few homemade remedies to Aphids and Mildew.