As others have mentioned, there are many different British accents. I wouldn't recommend a Cockney accent. However, a nice neutral British accent, like a BBC newscaster is a very good accent to have. Most Americans perceive people with such accents as being more intelligent and more attractive than people with neutral American accents. As someone who grew up in the Midwest and intentionally made my accent more neutral American while living on the East Coast, I have a pretty neutral American accent. However, now that I'm living in Hong Kong, I'm intentionally trying to pronounce my vowels in a slightly more British manner. However, I'm too attached to my Rs to switch to a non-rhotic British accent.

A slight Southern American accent sometimes does cause many Americans to perceive someone as more trustworthy, and some Americans do find a slight Southern accent to be very attractive. Though, be warned that a Southern accent may cause some Americans to perceive you as more stubborn and slightly less intelligent.