And whether the ship and its crew are actually better than others - their reputation is sure to have an effect on other crews.
Think less in Federation terms, maybe, but perhaps in Klingon ones.
Imagine you're the commander of a bird of prey, and on the losing end of a battle, when suddenly General Martok's cruiser arrives at the edge of the battle, blasting away at the enemies. The crew cheers, maybe breaks out into a war song (okay, that might or might not happen, but imagining is fun) and the desperate Klingon forces, reinspired, regroup for another assault.
True, Federation crews are at a pretty high standard altogether, but I think even among them are more and less experienced crews (and captains) which will react positively to a "famous" vessel in the area.
EDIT: And I DO believe the Enterprise holds the best crew in Starfleet. She's the flagship, and one of the most advanced ships in the fleet, and she'll be in some most important missions, why not crew her with the best they've got?
Or having some Admiral's ship in the area, acting as a command ship, giving orders to the fleet to use the resources at hand most effectively will have a similar effect. Yes, it could even be built into a whole class of 'command cruisers', as which Galaxy, Sovereign, D'deridex, Negh'Var and similar classes often act.
EDIT: And Baron is certainly right that the crew is important, not the ship, but it's MUCH easier that way. and "I'm Captain Picard of the USS Florence" or whatever doesn't have the same ring to it.
MORE EDIT: Maybe they'll just do it like with the Enterprise-A or the Defiant... take some ship of the same class, spray-paint a new name and registry on it, voilá, we have our hero ship back. (Or, in FleetOps: just build/call in a new one) And we can assume that even if the ship is destroyed, our her crew always gets out of there in time...