Because the structural elements are already in place, framing in an unfishined basement doesn't require structural walls or studs. As with exterior studs in residences, the size of the exterior studs is largely dictated not by strength requirements but rather by the depth of the bay. A framing bay from 2-by-6-inch studs holds more insulation than 2-by-4-inch studs, increasing the highest potential R-value, which measures a property's insulation. 2-by-2s will work just fine for exterior walls. If you have a greater insulation need than afforded by 2-by-2s, then 2-by-4s will work for your exterior walls.
If you make any structural changes, you will shift the load carried by the existing structure. Expanding a basement window-well for increased light and ease of egress, for example, may create a structural change. An exterior wall that may have been finished in 2-by-2s may now need structural framing. You'll need the help of a designer or engineer to make sure your new framing picks up the load shifted by any changes.
This is counterintuitive: when finishing basements, exterior walls may be smaller (2-by-2 studs), but interior walls need to be 2-by-4 studs---unless they're structural, in which case they need to be engineered. Even a non-load-bearing interior should be 2 by 4 inches. Another common use of studs when finishing basements is soffit framing. Soffits are framed down to cover ductwork and plumbing, and sometimes to provide room for recessed ceiling lights. As with interior walls, soffit framing should be done using standard 2-by-4s.
The proper way to address moisture in a basement is to first drain water away from the foundation. Second, it's to have a vapor barrier on the outside of the walls, between the soil and concrete or other wall material. You can't install a vapor barrier in an existing basement without excavating the perimeter of the house (which is done occasionally). So, sometimes vapor barriers are used on the interior of the wall, between framing and concrete. If there is a chance of intermittent moisture on the wall surface, pressure-treated exterior studs can be used.