Why do some commercial aircraft have no landing gear cover?
Yes, it does increase drag.  Anything that is not smooth, where the air flow will be disturbed, will generate drag.

The 737 was originally designed only for short haul and space is at a premium.  Adding doors and all the associated mechanisms adds complexity, weight, additional maintenance requirements and more possibilities for failure.

Boeing have had many opportunities to add doors and have chosen not to; so they, and their customers, do not see the drag penalty as more expensive than the extra weight and maintenance.