I've got a hanging basket that gets drenched whenever we have a downpour like the other night and rain overflows the gutters. A stream of water runs into the basket and right through it. Will that leach out any important elements from the potting soil? Do I need to over-fertilize it? Thanks!

Yes, water will leach out the nutrients in the potting soil, and an overabundance will do so even faster. Overfertilizing is not the answer, perhaps more frequent is. A very wet situation not only leaches out nutrients, but oxygen also. Your plants could drown. If the basket is subjected to frequent downpours, I would move it to another location.
Carole
JOCO Extension Master Gardener