I am building a laundry room in the basement. Do I need a drain?
You will need a standpipe for the washer to drain into, and a floor drain.  You will also want to make sure the finished floor slopes slightly towards the drain.

The standpipe should be obvious - if you don't have that the washer will just shoot water out the back when it drains.  Make sure it is vented properly.

You need the floor drain for when the washer leaks.  Washers can leak for a lot of reasons and they can dump a lot of water quickly - you don't want that water finding its way under the flooring, into the walls, and into any adjacent rooms.