Well, the way I see it is if they say they don't want or need help, no harm in asking and then going on your way. But I'd rather make sure than to ride by and leave someone who might be in need of help without even asking.

On my century ride a couple weeks ago, I came across a group of riders I knew, one of them was busy patching a tire. I offered my assistance even though I figured they had things under control, being more experienced riders than me. They said they were fine so I didn't insist, just stuck around and chatted for a while before pressing on. But yeah, it would have been a different story had one of them been pushing their bike.

And I will usually turn down offers of help because I have a spare tube and patch kit and am capable of fixing a flat on my own, but if they want to hang around and help or even offer moral support I won't turn them away.