Are you performing the work or is someone else performing the work?

Air sealing a gap like the one shown would typically involve either caulk or spray foam insulation depending on the width of the gap.

Fill the gap with DAP Dynaflex 230 Premium Elastomeric Sealant, Cedar Tan

Show the exterior of the door and frame. Any caulking squeezed into the gap from the inside will be a mess on the outside--and not colour matched.

Backer rod and caulk from the interior. Same from the exterior but color matched caulk as worthy alluded to.

We do interior air sealing with caulk which is required in the municipalities in which we work. It doesn't fill the void volume and it doesn't substitute for caulking and sealing on the exterior to keep the weather out. I would expect brick mould or other exterior trim to be installed on the exterior which would overlap the jamb. The light getting through suggests it hasn't been installed yet.

As mentioned above, spray foam gap fillers can be used to fill voids between the interior and exterior caulk.

For instance.

What about the other gap??