Oho! I think I can help. Against diddies I usually play as aggressively as I can so what they don't pull out bananas. If a diddy pulls out a banana, I'd try to grab it before he can, and then proceed to throw it away from them.
Note that I didn't say to throw them off the stage. What you need to do, and what I'd recommend to most characters playing against a diddy, is to throw them to where diddy either can't access it or must spend time trying to get it.
This includes the opposite side of FD, smashville's moving platform, and support ghosts. Any difficult-to-access platform works. Basically, if the diddy has to double-jump of cover a lot of ground to reach the banana, that's good. Also, you should try to get yourself in between diddy and the banana whenever possible.
This will force the diddy to go to some lengths to get the banana back, pull out another one if he can (which, if you're close to him, can be caught before he can do a thing about it), or just fight without bananas. Oftentimes you can hit a diddy a few times as he tries to reach his lost banana; if you're good, you can get him off the stage or at least continue to keep him away from his banana.
Throwing it at him is not a very good idea. He can simply re-catch it, or dash attack to catch it and hit you in the process. I'll wager that the diddy player will know how to use bananas much better than you, too. The only instance where I'd say "throw it at him" is if the diddy's offstage, where it could potentially mess up his recovery and where he can't pull bananas back out.
Though you can always catch a banana diddy pulls out before he can react, make sure you catch it quickly (while it's in the air); he'll peg you with something if you're too slow.
Um... basically, stay close to him and don't go high into the air too much; that gives him tons of time to pull naners out and get ready to punish your landing.
Be ever vigilant; the matchup takes time to learn, but once you learn a few things about how diddy plays, you'll be able to play against diddies competently with almost any character.