Foam insulation is definitely the right call for basements, and everyone here is spot on about the moisture issue being your real problem. That corner won't magically dry out just because you add insulation - it'll trap moisture and breed mold behind your walls. Get down there and trace where the water's actually coming from first. Could be gutters, grading around the foundation, or interior seepage. I had a similar situation a few years back and discovered my downspout was essentially creating a small river right at the foundation line. Once I redirected that, the dampness cleared up.
That said, assuming you can actually get the water source under control, spray foam or rigid foam panels both work well because they don't absorb water like fiberglass does. Spray foam gives you better sealing and air-tightness, which is nice. Rigid foam is easier to install yourself and less messy. With either option, you'll want a vapor barrier on the warm side of the insulation (the interior), but honestly if you're dealing with active moisture, the barrier only matters once you've fixed the underlying problem. If that corner stays damp, no barrier will save you.