How can I reuse lumber from the basement walls of my 1938 home?
You could certainly double the plates if the size difference won't be a problem, or you can solve that in one of several ways if it is. Don't use studs that aren't full height otherwise. They won't be stable at the center joints. 

If you're building new exterior walls, just flush everything to the inside. If not, you'll probably need to fur out at least the lowest and uppermost plates to flush on the other side as well. This will prevent cracked drywall, etc. Depending on the difference, 1/8" hardboard or 1/4" luan plywood could work well. 

I wouldn't worry about cutting off nail penetrations unless there's obvious splintering or other damage. It won't affect the outcome of your project. You might find it appropriate to square up the ends and make the lengths uniform, though, especially if they were hand-cut. 

Also be sure to crown them. This involves sighting down an edge from end to end and aligning all the bows the same direction (perpendicular to the wall). Discard any studs with bows greater than about 1/4", and use them for shorter lumber needs or where the bowing won't be a problem, such as in drywall backing.