Addendum: In some cases, we do use 'droog' as an adjective with a different meaning than 'dry'. However, it is true that 'boring' is not a meaning amongst these. :)
"Hij heeft droge humor." = "His (sense of) humor is sarcastic."
Sarcastic might not be the right word here. When someone's humor is 'droog' s/he jokes around without showing much emotion. S/he gives surprising and funny comments. In this respect, 'droog' definitely doesn't adopt the meaning of 'boring', as you can see. :)
"Het is hoog en droog." = It is 'high and dry'. & "Op het droge zijn." = "Being safe" (literally: being on what is dry').
In these cases 'droog' has something to do with being safe, with safety. For a nation always at risk of being flooded, it's only natural that these proverbs developed. ;)
But there is the proverb in German "seine Schäfchen ins Trockene bringen", literally to bring one's own little sheep into dryness, i.e. to get one's wealth to safety, which seems to be nearer to the Dutch meaning; probably because sheep are used to keep the grass short on dykes. Whereas "auf dem Trockenen sitzen", to sit on the dry, has a more negative meaning, usually just of an empty glass but sometimes even being bankrupt.
I think those old dudes are guilty of telling "lame jokes" (which are not humorous). "Dry humor," on the other hand, tends to be more sarcastic, or sometimes droll (I think there are several Jane Austen side characters who demonstrate this, in pretending to not be listening to the gossip around them, but then pipe in with a comment from time-to-time which is usually of the dry-humor sort).
I wonder if it's like English, where 'dry' is a certain type of boring, usually 'technical or complex in a way that discourages interest.' You wouldn't say "Waiting rooms are so dry because there's nothing to do," but you could say 'I thought a seminar on fantasy sports would be exciting, but instead it was a long, dry lecture on statistical analysis.'