Say you won a Nobel Prize, Fields Medal, or Presidential Medal of Freedom, or Congressional Medal of Honor, or Teacher of the Year - and it is only to be presented by Donald Trump (or Sarah Palin, or some other person you hate). There is no other way to receive the award other than to have this person present it. Would you do it? You do not get to make any speeches, etc. Or, perhaps your team won the Super Bowl and is now going to visit the White House.
IMHO, I cannot see how or why the answer should be no. If you like Trump, then what could be better than getting a high honor from the president you like? And if you dislike Trump, then wouldn’t it be ‘befitting’ to get a high honor presented to you by someone you dislike?
You decline the award to make a statement. That statement will likely get more press than accepting the award and therefore make your statement more powerful.
For example, there are many Best Actor Oscar winners, but I can’t name many/most of them. However, I can for certain tell you that Marlon Brando declined that Oscar because of the treatment of Native Americans in Hollywood (or in the U.S. for that matter).
Also, if the person you hate has a big ego (or maybe even not so big of an ego), the snub will be infuriating to them.
Piss on 'em, and piss on the award if the only way I can get it is by being in the presence of someone I loathe for the “theater” of the presentation. I will still know I won it, no matter what they do with it after my refusal.
I’ll point out that if you’re in the circumstances embodied in the highlighted phrases, you’re not in a position to snub the Commander-in-Chief. (Not without notable consequences, anyway.)
Decline the reward - particularly if it’s just an award. (Which is to say, this would be a much more painful decision if I was turning down a giant pile of money.) When the dust settles I’ll still know I earned the award - that’s as good to me as having some tschotske to put on a shelf. And knowing that I snubbed the egomaniac is priceless (as long as there’s no actual price).
It would have to be someone has bad as Trump for me to decline. I was no fan of G.W.Bush, for example, but if I won the World Series I’d show up at the White House and be polite. If I won this year, no way in hell. Not even something I’d have to think about. These are not normal times.
Does the award come with money?
I would decline unless there’s a significant amount of cash involved.
If the award itself was non-controversial, and Trump’s role was limited to handing me the trophy and shaking my hand, I’d probably show up. The “statement” I’d be making by declining wouldn’t be worth the blowback.
I’d be much more likely to decline if the award was tainted in some way—like if I won a prize for inventing some kind of tranquilizer, and Bill Cosby was presenting it. (Sorry, I know that example is really farfetched, but it’s the best I can come up with at the moment.)
I don’t understand the bolded part. How would it be “befitting” to get the award from someone I dislike?
North Carolina has a superintendent in the mold of Betsy DeVos. If he came to our school for a photo-op, even if he were presenting me with an award, I’d not want to be in it.
When a politician presents an award, they don’t do it for your benefit: they do it so they can appear in a photo-op with a grateful constituent. I wouldn’t want to be part of that sort of advertisement for someone whose political career I didn’t intend to support.
I think he meant that Trump would have to endure the humiliation of presenting an award to a political enemy. But, unless you’re appearing regularly on the news shows he watches, or you’ve gotten his attention on Twitter, he probably wouldn’t even know you’re a political enemy. And if he did know, he’d probably come up with some excuse to avoid the award ceremony entirely.
Yes I would definitely accept, or accept any reason to meet Donald Trump, if only to also ask him to sign this photograph.
Say my local Parks Board president is the person who routinely hands out the awards honoring those who volunteer for cleaning up our parks. Somehow I organized a bunch of people and did a good job, and my name came up as the guy who should be recognized this year. Even if I know he’s a drunkard who still beats his wife 
, I’ll accept that award because I worked hard, and the ceremony will make some small number of people aware of the problem of keeping the parks clean. The award is not coming from him, personally, it’s coming from whoever is the current holder of the office. It’s the office that matters, not the person. It seems like a kind of latter-day Donatism to think otherwise.
If the award means I’ll never have to work again, I’ll take it happily.
If the award means doodily squat to me, then I won’t bother accepting it.
If it is somewhere in between those two extremes, then it all depends on the award and the person.
The honor is already mine. Whatever process picked me has already happened, and my winning is recorded. Unless there is something substantial accompanying the certificate, I’m declining if it means giving any legitimacy to someone I detest.
I’ll take the award. I’ll be civil before, during and immediately after the ceremony.
But my FB and Twitter would have already made my feelings about Trump known, should he be the presenter, and I wouldn’t go through any effort to hide that or dial it back after the fact.
So Trump is a Hydra operative? I find my estimation of Hydra somewhat lowered. 
Don’t care about the money; I would decline the award. In a way (and much down the scale in terms of awards) I did. I didn’t care for who was handing them out so I respectfully requested it just be sent to me and allow me to bypass the Rubber Chicken Dinner and presentation; should that not be acceptable please pass it along to First Runner-up. I got the box in the mail.
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I would accept the award and be gracious about it. Even if ol’ Hillary was handing it out.