I've seen this happen when any cold surface is exposed to the elements when the humidity is high. This typically can happen on cold water pipes too, especially in summer when my AC was running. The only two practical solutions I'm aware of are 1. reduce the humidity, or 2. insulate the colder surface so it's not exposed.
I would personally prefer the latter option because it will work regardless if the power goes out, doesn't consume electricity and isn't as expensive as a newer dehumidifier. Now, if the rest of the basement is damp or mildewy or you have other mositure related problems, then the demidifier is probably the better option.