If it's an old house with good concrete foundation walls and good drainage, and hasn't leaked yet, you should be safe.

Another point - be sure that the basement wall vapour barrier plastic is sealed to the insulation's vapour barrier in between the joists against the exterior wall. (back in the 50's or before, vapour barriers were an occasional concept, usually that paper-backed fiber insulation was the vapour barrier. Proper insulation was also often just a suggestion.)

Also, by code - while pink fiberglas insulation can be exposed (with a plastic vapour barrier) - Styrofoam insulation must be covered with drywall; otherwise, a fire could burn the Styrofoam to produce noxious fumes. Sealing it behind drywall limits this risk. Also, know your code - electrical outlets in a finished room must be no less than 12 feet apart (since many appliances have 6 foot cords) not counting doors and windows. My local code allows for zero outlets in unfinished areas. (i.e. no drywall).

And... finishing a basement (or part) can add to your tax bill if the municipality gets wind of it.