Would it have wise for Robb to send Jaime to Winterfell under guard to prevent the risk of losing his most valuable hostage? Setting aside the benefit of hindsight (Robb couldn’t have known Winterfell would fall), it seems smart to keep Jaime somewhere more secure and under proper guard – it scuppers any escape/rescue attempts if he’s in a castle cell two whole kingdoms away from any Lannister forces, and Rickard Karstark probably wouldn’t have been able to try and kill him either.

Riverrun was secure enough; the problems Edmure ran into was that Tyrion was massively violating the social contract (and he did manage to prevent that prison break), and that he couldn’t have foreseen Catelyn’s actions either. 

Moreover, Robb doesn’t necessarily want Jaime in a castle cell two whole kingdoms while he’s got him hostage. Contrary to the fandom impression that Robb was completely opposed to peace with the Lannisters, Robb was willing to make peace on his terms and would have needed Jaime on hand if necessary to cement the treaty (even if that was a few years later).