As long as the floor doesn't have to carry heavy loads, (another wall, heavy appliances, etc) I believe you could get by with 2x10's on 12 inch centers. Cross bracing would be required, and I would do it at the center, and 4 feet either side of center.
In order to install sistering, cross bracing, etc, you will have to remove the ceiling in the basement anyway. So you might as well have at it to determine what you have for floor joists. If the floor is sagging as bad as you indicate, I would guess that you may only have 2x8's now. You could sister 2x10's to the 2x8's (if that's what you have)on 16 inch centers, and that would also make a significant improvement. The cross bracing (some call it bridging) would still be required. The 2x10's would also reduce the ceiling height in the basement by 2 inches if that is a concern.