Welcome Dooley, I love R+L=J threads! I was very skeptical of this theory myself but after combing through this forum and re-reading through the book with this theory in mind, you find a great many correlations and future possibilities involving this development! Forgive me if this gets longwinded...
----Warning! Spoilers!-----
Anyways, to answer your major question, there are actually two highly influential characters who know about Jon's true lineage. As the theory goes, Ned comes upon the Tower of Joy in the midst of Lyanna's birthing of Jon. To get inside, Ned and his companions had to defeat the three Kingsguard at its base in mortal combat. As described in the dream, only two survived. Eddard and his most trusted companion, Howland Reed.
Who is Howland Reed again? He is lord of Greywater Watch, bannerman to House Stark. He is described as one of Eddard's most staunchest companions during the war against the Targaryens by Maester Luwin. Luwin found it extremely odd that Howland did not show up to swear his fealty to Robb during his insurrection against the Lannisters. Instead he sent his two children, presumably to guide Bran to the children of the forest north of the wall. I believe Howland is one of the few who knows of the true threat coming from north of the wall. Howland (presumably) has been holed up in Greywater Watch, deep in the bogs of The Neck throughout the conflicts raging throughout Westeros. In fact, that is precisely where the remnants of Robb's army has fled to, attempting in desperation to enlist Howland's aid to take Moat Calin. Once the Red Wedding occurred it cut them off, and nothing has been heard from them. (That I remember at least) If the remnants of Robb's army hears of a true Stark in the north, would they rally around him?
Regardless, C of K is correct, if the Lannisters massacre everyone in the Neck who's to know? However this is still one other who knows of Jon's true parentage, and he is in an even more influential position than a noble of the North. This clue can yet again be derived from Eddards dream. The bantering between the Kingsguard and Eddard goes to great pains to highlight where each member of the Kingsguard was during the rebellion, (and where they should have been). Every group of them was guarding one of royal blood. The Kingsguard are also sworn to keep the secrets of the House Targaryen. The only one left of those original seven, Ser Barristan the Bold, would be able to derive this fact if he were not literally informed by Rhaegar himself when fighting beside him in the Trident. If he told him himself, then Ser Barristan would also know the reason why the three most prominent Kingsguard were escorting Lyanna to Dorne: Because Rheagar believed Jon was the Prince that Was Promised and thus needed more protection than Viserys, Aerys or young Aegon. Ser Barristan is now Dany's right hand man, and even offers to tell Dany the secrets kept for House Targaryen's ears only, but he is cut off until a more opportune time. One of those secrets, is likely Jon.
Whether TPTWP is Jon or Dany, only time will tell.
There's actually a third person who I think knows about Jon's parentage, but it wouldn't be a very accurate testimony. One of Bran's dreams was described in afterthought. He had a conversation with his father about Jon. Eddard seemed sad to Bran, when talking about Jon, and it was extremely unsettling to him. I actually forget the details, but it seemed like Eddard was trying to tell him about Jon beyond the grave, but Bran didn't like the idea of Jon not being his brother.
And if nobody other than Ned knew, who did Ned trust enough to tell? Benjen maybe? But if that really was the case, why would Benjen want Jon to join the Night's Watch, surely he'd know that he'd be destined for greatness? or maybe it was a protective measure.
I don't think even Benjen knew, I think Ned kept his secret close to his heart. I don't he knew about the prophecy either, he was sent to the wall to be as far away from King's Landing as possible, a place where even bastards can rise high.
Nobody would believe it, and what type of problems would that create between Robert's supporters, and Stark followers when it turns out that Lyanna was an adulterer with the ENEMY.
Robert's and Stark's followers are all but dead. As to the ones that are left, I think they'd turn to anyone whom they think could save them from the Lannisters' and Bolton's wrath. I think Targaryen stability will look pretty attractive after the harsh winter to come.
