My stair treads are shallow and people have fallen on my stairs. I would like to remove the carpet and install hardwood thereby making the tread deeper. The stairs originally led to an unfinished basement which has been finished and the stairs carpeted. Underneath the carpet the stairs are very rustic. Can I remove the treads, use the original stringers and install hardwood?

You can, but they wont be any wider. Basically you'll end up with the same dimension you have now. ( When you add overhang to one step, it takes away from the next step ).