The "basis" of an asset is what a person has paid for an asset together with money invested in an asset after purchase This determines gain/loss for income tax purposes.
A "stepped-down" basis is the same as the "stepped-up" basis - except the asset has gone down in value since it was purchased. The new owner would take that asset with a new lower (stepped-down) basis. This depends on the value as of the date-of-death of the owner compared to the price the owner originally purchased the asset.
Example: Dad was convinced that AOL was going to make a comeback. He bought it for $50,000 and it’s now worth $10,000. Instead of selling his AOL stock at a loss during his lifetime he held onto it. I now own it with a basis of $10,000.