Build a plywood box around it. Line the box with carpet or some other soft
material. Leave a few decent sized (3" ?) holes in the box to provide
airflow.


That would help reduce the noise, however it could create a hazard.
That could cause the unit to overheat, also it would not have adequate
air circulation to dehumidify.
-udarrell

In addition to potentially overheating, it seems like it just wouldn't
work as well, unless it had a really strong fan to circulate lots of
humid air in through those 3" holes. The whole point is that it needs
to pull in the humid air, pass it over cooling coils, and blow out dry
air, which would not work well in a box. The air inside the box sure
would be dry, but I have doubts about the rest of the basement.
My advice: if it's a specific part of the dehumidifier that's
rattling/vibrating, see if you can glue/clamp/rubberband/tie that part
so it doesn't vibrate. If it's really the motor making all the noise,
there's a chance that bearings could be bad or the fan blades could be
unbalanced or something like that, but unless you're a confident DIYer,
it would probably be more expensive to fix than it's worth. Try a new
dehumidifier - pay attention to the return policy and try it for a
week.
Good luck,
Andy