Usually because the basement was initially unfinished, and the homeowner wouldn't want to look down the stairs and see the raw concrete floor and unfinished walls.  It probably also shielded you from the noise of the washing machine and furnace.  Unfinished basements can also smell a little dank.

adding to what geezer says:

There could be an energy code component to the thing also, depending on how the envelope is insulated.

There is no fire code issue, like a garage however.

heating. 

Rats climb stairs?

Have fun hauling the drywall downstairs...build some neat 3x3 wood panels in the garage and have your wife carry them down, a lot easier.

"rats climb stairs" - Finishing the basement has been part of the plan for  awhile now, but a couple weeks ago our dogs found a chipmunk in the basement and although I was able to rescue it mostly unharmed, I had to spend a good amount of time sealing small holes along the sill plate.  That task has evolved into insulating the joists, which turned into full insulation, elect, drywall, etc.

I originally wanted to remove the door to make getting the drywall in way easier.  Looks like it will go away permanently based on what everyone's said here.

Actually thinking about using small plywood panels for the ceiling (easy to take down if needed, screw directly to wood trusses)

Most of the time a door helps prevent cool air from your AC settling in your basement during the summer months. That is why it is preferred to have your door at the top of you stairs leading to the basement. Remove that door and you will notice that your AC will run quite a bit more than if you had a door. You should save quite a bit more on energy cost if you keep a door.