widening a narrow stairway going to the basement

I live in a two story row home that is a total of 13 feet wide. i just purchased a new washer from Lowes but when it was delivered it did not fit down the stairway into my basement which at its narrowest is about 25 inches wide. The washer was basically the smallest (non-portable washer) they sold. In looking at options for a washer that will fit i found that i would have to pay at least $250 more than i was looking to spend. So to my question.. my stairs from then first to the second floor runs along the party wall between me and my neighbors house and below the stairs on the other side is a 2x4 wall with a door leading to the basement. I'd like to knock out this will and door (potentially creating just a half wall) so that ill have a more open concept living area as well as more room to get the washer i want, plus coaches and such down into my finished basement. Are these type of walls typically load bearing? If not is this something i could potentially do myself or have someone do for around $250-$500?