Everybody is different, but I walk. If I stop and stand, the muscles in my legs won't continue to pump and keep the circulation going, otherwise everything pools in my feet. For long straight downhills where I don't care about gaining speed, I'll even keep my legs gently turning over so that circulation and flushing is maintained. If I'm riding a familiar route and know that a particular hill is going to be a problem, I try to get my heart rate down by riding at a very low effort for quite a while before.
Good recovery after a ride includes putting your feet up, not standing still, so it just seems that walking is better than standing. Only time I stand is when I've stopped at the flattest place and couldn't get started elsewhere. I kind of view it as a mini-cooldown period like you should do at the end of a ride.
There is no rule that you have to ride the whole thing. In a mtnbike or cyclocross race, the racers take the most efficient means of locomotion. If that means getting off the bike and running up a hill with it, fine.
Fitness, gearing and confidence will help you conquer the hill one day.
Frends know gud humors when dey is hear it. ~ Da Crockydiles of ZZE.