Dove's Dual Role
Since Hawk does much of the heavy hitting (literally) in the Hawk & Dove comic, it could easily be argued that Dove isn't a necessary character. I beg to differ. I believe Dove actually performs two roles in the comic storyline: firstly, she has to restrain Hawk from going overboard (he is the Avatar of War and derives his power from Chaos, after all); secondly, she must step in and save the day in small, humble ways that may not get a whole lot of attention.
Her First Job: Keep Hawk From Going Nuts
Dove's first priority--at least, according to her actions--seems to be keeping Hawk from getting into too much mischief. After Don's death and before Dawn came along, Hank acted as a laughable vigilante, trying (and failing) to fight crime as he and his brother did. After Dawn comes into his life, Hank is actually doing some good as Hawk, even though he's still pretty reckless. Dawn's presence, much like Don's, helps Hank harness all that energy and make it more productive.
Her Second Job: Preserve the Peace
While Hawk is usually smashing stuff and going toe-to-toe with big bad guys, Dove's usually using her super-awareness and deft grace to keep the worst from happening in the background. Whether it's outmaneuvering a bad guy, or keeping innocent civilians from being hurt, Dove is usually the one taking care of such smaller heroic tasks, making Hawk's job a little easier and more straightforward.
Conclusion: She's Invaluable!
I believe that both of these roles Dove plays, while not as traditionally powerful as many superheroes, make her an invaluable character, both within the context of the comic and outside it. After all, those who do what needs to be done without saying much about it should be honored, too!