Well, we've seen how Ron reacts when family members are hurt (eg. Arthur being attacked by the snake it OotP) - he seems to act quiet, shocked, brooding. So you can apply that to this situation. He'd be scared but trying not to show it too much - trying to be strong. If they were kidnapped or something, he'd be angry, enraged. He's probably pretty possessive and protective of his kids.
He'd definitely swallow his pride and ask Harry for help if he felt his kids were in danger. The first person he'd enlist to help would be Hermione, though, of course - but if there was danger, then he'd be trying to protect her, too. He'd probably rather make sure she's safe and get Harry to help him. Whether Hermione would let him go off to save their kids without her is a different matter ;)
Hope that helps!