Don't just screw them to the sides of the existing ones, and don't make the new ones bear more weight than the existing ones by lifting the floor with a floor jack before you screw them in. The new 2x's need to be just as long as the old ones, just insert one into the middle of each bay (if the old 2x's were 16" oc, now you will have "joists" every 8".)
The new 2x's need to sit on the top plate of the downstairs walls on both ends, just like the old ones do.
As far as the cross bracing steve mentioned, that is purely to prevent the joists from falling over, and the cross bracing was rarely installed in those days in such a way to actually prevent this. A solid block of wood in the bays is far more effective and what is more commonly and easily framed now. (ie, if the floor joists are 2x8's 16" oc, then you would see a 14.5" piece of 2x8 sitting perpendicular to the joists in every bay, alternating every other bay so it can be nailed in, but that's a discussion for a different question.)