Honestly ... most of the time ... we don't. Just as we don't usually think of any other country, or its people ... unless it's in the news. But when we DO think of the British ... we tend to feel an affinity for them. Unless we're decendants of Irish immigrants. lol
After all, England is the "mother" country to America. And if the colonies had been treated right, we'd still be subjects of the crown.
Basically, I feel that Americans appreciate the British (specifically the UK) as a staunch allied country. But as far as their culture is concerned, we most likely think of them as a bit stiff, somewhat wussey and far too obsessed with useless royalty. And maybe just a tad dumb, since they ushered in a national health care program that is causing them to drop like flies.
Is quite offensive saying that our monarchy is irrelevant and should be dethroned. Our monarchy is part of our national identity just as you have your president. Us Brits don't go mouthing off about your president. Have some common decency for god's sake