My neighbors have a dirt and rock yard and the run-off floods a shed. How can I inhibit this?
You can place a french drain on that side of the shed.

The other thing you should do is to find better material for those sole plates - I am thinking they are 4 x 4 all around - so you might try something like "Plastic Lumber" - there are many composites out there as well. This will eliminate your issues.   You can search for plastic lumber or composite wood 4 x 4  as well..

The real issue is as one commenter stated the runoff from your neighbors - I see a chicken wire fence (perhaps chickens - if this is in the county and not the city - you may not get the support to resolve the issue form your neighbor, as the requirements may be much more lax.