Killie are working with Police Scotland to identify individuals involved in the letter sent to the club in the wake of the 2-1 home defeat to Livingston on Boxing Day.
Motherwell manager Robinson, whose side host the Ayrshire outfit in the Scottish Premiership on Wednesday, described the abuse as "simply not acceptable" and assured Dyer that everyone in the game is supportive.
The Northern Irishman, who received a death threat to his children four years ago when he was Oldham manager, said: "I find it incomprehensible that people do that kind of stuff.
“It's the modern society we live in. Abuse in any form is not acceptable.
“I think in general football managers are seen as fair game for some reason. You can say what you want, do what you want and no punishment seems to happen in general and obviously the racist element is just not acceptable in this day and age.
We are going into 2021 and it would be nice to start the year with some positivity and not with the mindless idiots involved.
“It is 2021 we are going into not 1821 and I find some of the abuse unbelievable. Hopefully we can stamp it out.
“We fully support Alex and the football community is completely behind him, but so should society be.
“As I say, it seems to me you can say what you want about football managers, their families, abuse them, do what you want to do. But no form of racism or abuse is acceptable.
“We are going into 2021 and it would be nice to start the year with some positivity and not with the mindless idiots involved.”