Absolutely not!
Taking advice from an internet stranger is one thing, but entrusting em with repairing your electronics (whether by mailing it in, having them come over, or you meeting them) is a big no no in my honest opinion. You don't know that person, you don't know if they're trust worthy, and who knows if you'd get your board back (if you mail it) and at what state! And if going with one of the other routes mentioned (them coming over, or you coming over to them) like, again, you don't know that person (and what they're capable of!) what if they're some deranged violent psycho? IMO, it's better to be safe than sorry, and over-trusting strangers is a bad idea. You may disagree with me on this (and you're entitled to) but I 100% stand by my opinion.
Why? First this is a dishwasher control board, not a priceless antique, so what's the worst that could happen? You send it to someone who drops off the face of the earth and you have to buy a new one to replace it. The alternative if you don't know how to fix it yourself is to just go straight to the sourcing a replacement part. At worst you are out the cost and effort of mailing it out to somebody. There are a lot of people who have been on this forum for long enough to have quite a history of posting, I would not hesitate to trust the more established members. On top of all that, most people are decent, I've bought and sold items with complete strangers more times than I can count and I've never had a bad experience with it. I've done deals with members of this forum and others that involved buying and selling items where they one of us could have easily taken the money and run but nobody ever has. Back in my 20s when I was between jobs I even did some general handyman work, I posted on craigslist back when it was somewhat less sketchy than it is now, random people would contact me and I'd drive to their house to do stuff like replace a light fixture, fix a broken switch, un-jam the garbage disposal, fix the clothes dryer, etc. Never worried me in the slightest, most people are not evil and I never carried around a lot of cash or anything like that. I've also generally been able to get a pretty good reading on people by exchanging a few emails. It's totally silly to worry about, his request seems completely reasonable. Frankly you sound extremely paranoid.