At my parents, the basement would be quite humid in the summer. That's where my bedroom is/was and the washer and dryer. We didn't have problems with the washer/dryer smelling, but it's possible that if they haven't been used for a while they soaked up bad smells? Maybe after a couple uses they'll smell better? Definitely try the cleaning routines mentioned by the PP.
As for the dehumidifier, because I was sleeping in the basement, I wanted to be comfortable. I'd run the dehumidifier a couple hours here and there in the summer. I didn't use the continuous run/drain option, I just emptied the water basket as needed. Using the dehumidifier made a HUGE difference in the feel of the room. It was really necessary to run it during the summer. We bought a brand new one for about $200CAD? so I'm sure it cost less to run than an old one. I don't recall electric bills skyrocketing.
During the winter, we turn on the electric baseboard heaters in the basement. Turning on the heat solved the dampness problem.
ETA: A quick, easy way to close up air leaks and drafts is to buy a couple bottles of spray foam and sealant (caulking) to seal around doors and windows and any other place you feel a chill. Also think about covering windows with clear plastic (or even walls!) with heavy curtains, quilts or woven rugs. Have plenty of throw rugs on the floor where you stand/sit for a while (in front of kitchen sink, in front of sofa, beside bed etc)
Also make sure you have your basement heat on but set low so that your pipes won't freeze this winter. I think we have our basement set to 12C but I know the previous owner had it at 10C.
Blogging about renovations in our first home