Obviously, paying the store to deliver is the smart way to go. That way, you aren't responsible if your expensive new washer gets damaged. Okay, I get it...you're going to do it yourself. There are a couple of options. Renting an appliance dolly makes the most sense, but you can probably make do with your standard dolly (assuming it is rated for the weight). Your washer probably weighs a little less...but within spitting distance of 200 lbs. Depending on the method you choose, the weight may be reduced by friction, but worst case is that each of you will need to lift (or hold back) no more than 100lbs.
If you opt to use your regular dolly, you'll want to strap the washer to it (like on an appliance dolly). Then you get one person above, and one person below. Walk it down the stairs, one step at a time and take time to rest mid-way, if needed. Gravity will do the work. You just have to control it for 8 inches at a time. Make a few practice runs with the empty box to be sure you have the hang of going step by step, and that there is enough clearance. The last thing you need is to get stuck halfway!
The other method is to slide it. In fact, just last week my wife and I slid a new water heater into the basement. I got a long 2x10 and laid it along the steps as a slide. Then we laid it down at the balance point on the top step and both of us got below it to guide it down. Friction helps reduce the weight, so it's quite manageable. Best if it's still in the box, otherwise it'll get scratched up. Sounds like the box won't fit through the door, but you could at least lay cardboard under it.