How do I determine if my basement leak is a foundation problem
It's nearly impossible for us to diagnose this without being there in person, but typically wet basements are caused by ground water--rather than a leaky window or the like. 

The ground water needs to go somewhere. If the soil can't drain, it builds up pressure and looks for a way into your basement--either via the walls or, more often, where the wall connects to the footer at the floor. 

Cheaper/easier things to try first:


Add/fix gutters and downspouts 
run downspouts further away from house 
properly slope the ground away from the foundation around the house.
install surface drains (such as a dry river bed)
have gutters/surface drains go to dry wells


More expensive/complex solutions:


Install internal footer french drains and a sump pump
dig down exterior to install exterior footer french drains and waterproof exterior


Fixing gutters and such is fairly inexpensive and a weekend project. Sloping the yard and surface drains are a bit more work, but still in the realm of DIY.

Installing footer drains and waterproofing the exterior below grade, however, is not a DIY project and will cost quite a bit.