The white powder is lead sulfate caused by the reaction of leaking acid fumes and the battery posts and terminals. Air and moisture help the reaction. The most likely source of the leak is the seal between the post and the battery case.
Follow Mad Amos's instructions for cleaning the posts. Use hot water if you do not have baking soda. Rinse the posts again with fresh water. Lead sulfate is toxic, do not ingest or inhale it.
Coat the posts and terminals with a
sealer or
dielectric grease, available in most auto stores. Haynes repair manuals suggest using petroleum jelly (vasoline). I prefer the silicon based dielectric grease. It lasts longer.
There are also chemically treated battery terminal washers that claim to help stop corrosion. I have not used these so I can not comment on whether they work or not.
If the leak is large it will break down even the dielectric grease and it's time for a new battery.