It seems to me that many people put subfloors in their basements (and other floors), but what do they do when it comes to the existing stairs? I'm code says that you can't have more than 3/8" difference in the steps, and most subfloors are more than 3/8" thick (especially with flooring on top of it). What do most people do to fix their stairs?
Technically the floor below the last step is considered a landing, and stair risers may be measured from tread to tread, not landing to tread. The 3/8", depending on your exact local codes, may not apply to the tread to landing at the top and bottom of the stairs. It may have a different standard, or it may have the same standard. Of course if the height difference is large it's going to cause tripping and stumbling, but for minor changes a little more than the 3/8" code it may not make much of a difference or present a safety hazard.
However this can be fixed by putting a thin board on the bottom tread, then half that thickness on the next tread, then half that thickness on the next tread until the requirement is met. If your subfloor is under 3/4 inch higher than it used to be, then it should only take one 3/8 piece of wood on the bottom step to make the first two risers only 3/8 shorter than the rest of the risers, for instance.
You can, of course, replace the stairs or stringers entirely to meet the code requirements and account for the new floor height at the bottom landing. If the height difference is large enough that you'd have to modify more than half the stairs you'll probably have less work replacing the stringers than adding height to each tread.