Is sagging between the support posts? Is this beam visible in the basement? Are you sure the beam is sagging or is there other settling going on?
If your main beam is sagging between the support posts, then you will have to add additional support posts between the existing ones. You can do this if your basement (assuming you have one) as long as the floor is concrete and is on solid footing.
To minimize cracks, don't try to level the beam the entire 1 inch in one try. Do about 1/16 inch a day until you obtain the amount you want. This will minimize any cracking of the drywall, etc.
The amount of jacking you are considering will not cause structural damage.