probably a fungus attacking unfinished compost .. in other words.. doing their job further breaking down the compost ...

if so.. ignore them ..

what media did you use ..

ken

I have the basil growing in dirt, really well - in a planter attached to a rain barrel. It's worked for 2 years. When I pop the brown pod open...tiny white - what looks like seeds are inside

As was said, probably a fungus.
Look up bird's est fungus.
The "seeds" are small spore packets.
Not damaging to live plants.