Should I continue my basement subfloor under the stairs?
Practically speaking, it will be better to keep the integrity of the new floor intact. But the small unsealed gap will probably not be a major concern if you leave them on the slab.  I do have to say, however, a difference of 1 3/4" on the bottom riser does not meet current IRC code. The difference in riser height cannot exceed 3/8 inch. You may be grandfathered since the stairs already exist however. The only way to be compliant is to recut all the stringers adjusting the riser heights to stay within 3/8" max difference at any riser.