There are some sections within the movie where Michael Dukakis comes into view.
- The television
- The argument between Eddie and Elizabeth
What is the purpose of Dukakis and politics within Donnie Darko?
The appearance of the election seems to be just a representation of the times and possibly a direct parallel to the feelings of Donnie himself. An election represents uncertainty and balance shifts whereas Donnie's life seems to mirror that setting of an uncertain youth. Battling with bipolar disorder having his balance shifted a lot of the time combined with his new 'realization' that the world is going to end.
A simpler explanation that goes back to it being just a showing of the time era is the Father's obsession with Dukakis being shown that he is just a republican father and sarcastic person, the second also noted by his obvious sense of humor while the kids are fighting at the dinner table.
I remember from viewing with audio-commentary, that Kelly said in the first appearance of such a scene, that he didn't want to take any side. So I think he included it to narrow down the timeframe. There are many other indicators that show, that the movie takes place in the 80s. It seems normal, that in a time of an upcoming presidential election the candidates will be discussed (and will be shown in TV sometimes).
In the Donnie Darko Book, Richard Kelly talks about the dinner scene where the family is discussing politics. The parents are clearly Republicans but when the argument escalates and Donnie and his sister starts cussing at each other, the dad has no real problem with it and even finds it funny, even with their 9 y.o. daughter at the table.
The reason for that, according to Kelly, is that they have raised a liberal household and thus allowing their kids to be liberal with their politics.
Dukakis in this context is just a figure of their own household politics.
Kelly was raised in a conservative household and since most of Hollywood is liberal he did not want to "take sides" or push an ideology through his film.