Quick answer is - not a lot.
Diversion airfields aren't on the address list for a Flight Plan unless specifically requested by the operator, or unless it's manually addressed by someone in your departure airfields AIS who is feeling helpful. On that basis if someone picks an unsuitable alternate - that's their problem. They will only get told about the unsuitability if that alternate gets a copy of the plan
I know what you mean - I've seen this happen in the past when the airfield I've been at has been included on the address list (we were due to close an hour before they would even have got airborne!), and we've had to try and get a message to the pilot that we can't be his alternate. Taking a somewhat heartless line I'm afraid if they can't plan properly then they ask for whatever happens.