What could be causing my sink and dishwasher to back up?
It really sounds like the drain line blockage is beyond the place where the basement drain exits the foundation and goes off to either the public sewer system or private septic system. Could be broken underground line, line filled in with tree roots seeking water through joints or holes in the piping, or sediment / deposits in the line that have restricted its flow. 

Probably the best way to troubleshoot this type of problem is to use a inspection camera that is slid through the pipe on the end of a long cable with a camera display at the end. Some professional plumbers have these for sewer inspection. It may also be possible to rent one and use it yourself but then you do not get the experienced eye of the operator knowing what nuances to look for.