I have never seen any UI that depends on what hand a person prefers. I have never seen any options or profile settings for left-handled persons. Maybe it is a strange question, but I have to ask it. Is it common to skip these users? Maybe it is not so important for left-handed persons. In my opinion, I would like when some things (often used) are placed from the right hand (menu, card, shortcuts)
Only physical devices should be affected by whether the person using them is right handed or left handed, for example mice.
A touch screen is one area where you might want to consider such things, as you have considerations of the person's hand physically blocking the view.
When relating to regular site and applications, the right (pun intended) question to ask is what is the user's preference, OR take into consideration the orientation of the person's native language.
For example, most languages are written left to right, hence your UI should take into account that the user will view the elements on the screen in that order (and top to bottom).
If you do allow customizations, it still doesn't depend on whether the person uses their right or left hand. For example, a friend of mine uses Windows' task bar (where the "start" , quick launch and minimized applications are) docked left, instead of the regular bottom.
It makes more sense to ask what the user's preference is, i.e. do you prefer the viewing pane on the right or on the left, rather than asking them about being left or right handed and making UI assumptions based on that.
Somewhat related to Dan Barak's answer, I find it worth noting, that a touchscreen device - especially a mobile, one-hand operated touchscreen device - is not always operated by means of the user's dominant hand (Silfverberg et al. 2000). Approximately 93% of the population are right-handed.
Furthermore, there is a certain physical dimension not to be overlooked: Several studies (Perry & Hourcade 2008; Karlson et al. 2006; Parhi et al. 2006) have found that users prefer to interact near the center of the touchscreen, thus avoiding extreme flexion and extension of the thumb. (And to avoid total occlusion, one suspects.)
Karlson, A. K., B. B. Bederson & J.L. Contreras-Vidal (2006). “Studies in One-Handed Mobile Design: Habit, Desire and Agility”, Tech Report HCIL-2006-02, Computer Science Dept., University of Maryland
Parhi, P., A. K. Karlson & B. B. Bederson (2006). “Target Size Study for Thumb Use on Small Touchscreen Devices”, Proceedings of the 8th Conference on Human-Computer Interaction with Mobile Devices and Services, vol. 159, September, 2006, pp. 203-10
Perry, K. B. & J. P. Hourcade (2008). “Evaluating One Handed Thumb Tapping on Mobile Touchscreen Devices”, Proceedings of Graphics Interface, vol. 322, May, 2008, pp. 57-64
Silfverberg, M., I. S. MacKenzie & P. Korhonen (2000). “Predicting Text Entry Speed on Mobile Phones”, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, April 2000, pp. 9-16
My wife is left handed but she uses the mouse in her right hand. So do most of the left handed people I know.
Most computers are set up with the mouse on the right so instead of having to switch every time left handed people just get used to it.
We don't make left handed door knobs, or left handed stick shifts.
(As a bonus, left handed people are forced to be more ambidextrous and according to many studies this make them more creative)
Being right or left handed doesn't have much bearing on how you browse the web. I am right handed but I find it perfectly comfortable to start on the left of the screen and work my way right. The same would go for someone who is left handed. What matters more is not whether you move the mouse with the left or right hand, but what direction you read from. Westerners generally read left to right so that's what we give them.
Another important aspect is mental models. Like Sruly pointed out, many left handed people work many things right handed way. Surely, that develops a mental model.
When you think of UI, there are a few things which change based on what is the prominent hand for a user. One example is of keyboards, where in the traditional key layout, the thumb travel to hit enter or backspace for a left-handed person is significantly more. In these cases, you do offer options to flip the layout.
In many other cases, user get used to of the layout, and as other answer pointed out, somewhat ambidextrous people use interfaces as a right hand person would too. As their mental models are set to expect things like a right handed person would expect.