What should I do with insulation in basement ceiling?
Do not remove the insulation. You will degrade the overall effectiveness of the R-value (insulation rating) of the walls.

Continue the finish materials (it looks like wallboard) up into the cavities between the floor joists. You will need to install vertical nailing strips on the sides of the joists, and on on the underside of the subfloor would also be a good idea.

You can then tape the horizontal butt seam, sand and paint.