Spoilers ahead: I tried to look for a discussion about this in the forums and couldn't find it. If there is one just point me to it. Sorry and thanks.

If you don't get the dragonstone from BFB before you go to Whiterun, you get the quest from Farangar. When you come back Delphine is there. Well, when Irelith comes running in to report the dragon attack, why wouldn't Delphine run out with you? She's so into dragons and their history and the fear of new dragon sightings, why wouldn't she welcome this opportunity to run out there and help kill it? I followed her out of Dragonsreach but ran back in. I assumed she goes straight back to Riverwood. But if she did go with you she'd have seen first hand that you're the dragonborn. You wouldn't have to prove it later. Of course it doesn't break much immersion because you're not suppose to know who she really is at that point. But I'd sure like to know after she steals the horn.

Just a guess? Delphine knows that if anyone but a Dragonborn slays a dragon it will not stay dead, Alduin will just resurrect it so why should she bother? Besides she has other, more important fish to fry like finding the Dragonborn who can put an end to the dragon menace. Which adds up to this: Why should she put HER life on the line until she finds the DoVahKiin of the prophecies. Until you slay one of the dragons and make a name for yourself you are just another aspiring hero while she has to remain alive to keep the Baldes torch burning.

That's a good point Arnathorn. Thanks. I forgot the dragon won't die part. Maybe she thought Whiterun was doomed just then and figured she'd better get out of there. haha.