Is joist sag at connection with header a problem?
Good question Paul, The situation with your framing is not really normal, but not uncommon  for a house that age. Any time there is separation of framing members, it is a reason for concern and should be addressed. However, I would not say it is a major or alarming problem. There could be a couple of different reasons for this separation. The joists may have shrunk over the years from not being well dried at the time of construction. The other reason could be settling of the outside supporting structure or foundation. A good first step is to confirm level of the joists using a long level or a laser. I prefer using a laser as you can check level and deflection at the same time. This may also help you determine the cause of the floor sag. Any irregularities can be corrected by slight jacking, sistering, shimming or adding supports.

Assuming your framing is fairly level or corrections are made, I would recommend adding joist hangers to all the members. It would be impossible to close the existing gaps, and the gripping force of the existing nails has been compromised over the years from wood shrinkage. Simply hammering them in tighter would not be a permanent solution.  When selecting the hangers, be sure to get the ones that have holes that allow securing to both the joists and center beam. Use the appropriate size joist nails or screws for the holes in the hangers, don't under size and never use drywall screws.  If you use joist nails, get or rent a pneumatic palm nailer, it will make the process a whole lot easier.

Once you have properly installed hangers, you can go ahead with your finishing project being confident there will be no further separation of the joists from the beam.