How can I rephrase this so it's not a run-on?
but some are likely to object on the grounds that Putin is simply asking Americans to not attack Syria for genuine reasons
It's not really a run-on sentence. It's convoluted, but it's only one sentence, not two combined into one. Is there more of the sentence, another clause that precedes the "but"?
It's not clear who has legitimate reasons, Putin or the Americans.
"Some are likely to object that Putin is asking Americans not to attack Syria even if the Americans have legitimate reasons."
Omit the "but" at the beginning and it is a correct sentence.
Yes, this is the whole sentence "Those who make up Putin’s audience are so diverse in their views that it is hard to generalize about them, but some are likely to object on the grounds that Putin is simply asking Americans to not attack Syria for genuine reasons."
The whole sentence is awkward but grammatically correct.
To rephrase the sentence so it's not a run-on, you can break it into two separate sentences. One way to do it is as follows:
"However, some may object to this request from Putin, citing genuine reasons, and expressing their desire for Americans not to attack Syria."