building a new house, I beams and floor joists different height
The answer to your question is yes, you can drop the joist lower into the basement via different means depending on construction method.

Although typically in most basements the joist do not hang inside the foundation wall; they either sit on top of framed basement walls which can be easily lowered in height given basement walls typically have every stud cut to reconcile an uneven floor; or they sit on top of the foundation wall and the foundation wall would need dropped in the area of the deeper joist.

As a general rule, we avoid supporting things by ledgers if we have the option of sitting on top of something.

Another option that may not be of much expense is to raise the depth of the other joist to 11-7/8"; a plus is using a minimum strength 11-7/8" I-joist would give you a very stiff floor compared to the 2x10's.  It would also simplify the framing a bit to have a cohesive floor depth throughout the main level.