Might be slugs or snails - they have no problem climbing into baskets. They may hide under the baskets during the day or may be hiding on the soil under the leaves. It looks as though they've had a go at a leaf or too as well.

I think Stera is right. I also have nonstop begonias. When I planted them, I added Osmocote slow release fertilizer and slug/snail pellets. Not a nibble to be seen. This seems to work for me, give it a go, (Red & Yellow).