A low power hair dryer or set on low heat and monitored can be a help to get
air circulating. Damp plus no circulation can equal dry rot!
If the moisture has got into composite wood materail such as a strand board
floor or the cabinet itself? it may take a while to dry out!
Makes the point too that with almost all home repair it's best to get at or
at least temporarily alleviate the problem; for example a small bowl or
empty can placed under the drip and emptied every couple of days?
I was once involved with a public housing authority. Some homes had
conscientous occupants who would either fix small items temporarily or would
phone the maintenance office at first opportunity. Their homes were always
in the best shape!