How can I prevent mold on the wall/particle board behind the kitchen sink?
It's hard to tell from the picture but it looks like it was tile mounted to a wood board.  Wood is one of the worst materials for behind the sink.  If going to replace with tile, use a concrete backer board instead and consider using tile trim caps to cover the top.  Otherwise use a solid stone or stone like material.  

And make sure the caulking is in good shape.  Every couple of years it needs to be stripped out and replace.  If it is starting to look old and cracked, it's been there too long.  No mater what the material, water under or behind the splash with always cause problem.s