I think it depends. If there is some history of that name, then it would be good to use them. I also think if you use common names like "Sarah" or "Andrew" then you should use last names, because obviously there are more than one person named that. But if these people are in some civilization where the names are like "Edwalda" or "Derimia" and there is no significance in the name, it's your choice.
Last names are useful if your fantasy world has last names. If your fantasy world doesn't have last names, well...*shrug.* My fantasy world, loosely based on an American Indian culture, of course does not have surnames--so surnames are of no importance to those characters. Surnames are of use to the more European characters who come from cultures with surnames. It all depends. In one scene of my story, a modern-day American character, when speaking to a little girl named Winter Born (whose parents' names are Black Elk Horn and Silver Eagle Feather), tries to admonish the girl by using her full name the way modern-day parents do--and ends up saying, "Winter Born Eagle Feather Elk Horn!" because she has no clue how surnames work in this place or if they even have them. Of course, her comment makes no sense to the girl, whose culture doesn't use last names.
Depends on the situation. I usually give main characters last names, but not third-party characters. In formal settings, it's not uncommon for people to refer to one another as Mr. or Ms. Whatever. In my current work, the MC is often called by his surname alone, as are a few other characters and only close friends use first names. Of course this is all a matter of style and setting. If you don't plan on using it, don't bother with a last name. Tacking on unecessary names isn't characterization unless it means something.
The nice thing about adding last names is it gives you more diversity in dealing with characters. Without last names, you're limited to "Sarah" and a couple of pronouns like she and her. When you add the last name, other characters can refer to her as "Miss (last name)", full name or even just the last name. So, the three options of "Sarah, she and her" now grow to "Sarah, she, her, Miss Sarah, Miss Winslow, Sarah Winslow, Winslow" and other variations on the name that someone might call her. For the readers, it diminishes repetition.
Also a last name, or surname, shows which family you are from. Perhaps this is important because the McCoy's don't like the Clampett's. or what ever factions you have in your book. There are other ways to add a bit of depth and individuality to a character. you can say 'I am Bill'...Hi Bill. that's all well and good. You can also use Surnames like european style. Bill Thatcher. Or you can even us an older style like the Norse. 'I am Bill, son of Jack'. It can go even farther back. 'I am Bill, Son of Jack, Son of Tom.' How far it goes back depends on how far you want it to. just an idea.
In dialogue. If you start playing around with your character's names in the narrative it'll confuse things. It's considered bad writing.
If the name's origin/meaning is significant, or it just adds depth and interest to a character, then use it. I wouldn't use last names for every character, though.
Traditionally, surnames (last names) are used to address adult males & personal names (first names) address women & children. But as a writer, you create your own traditions.
If you are writing about different worlds, time dimensions, whatever, you don't need to be tramelled by traditional Western ideas. For example, in Turkey, we didn't even HAVE surnames until June 1934. (This means that some older women here haven't got 'maiden names'--it causes loads of confusion with visa applications!) We still don't use surnames much except in parliament or for formal identification.This is why our footballers, if you ever watch European football, have their first names on their shirts. There is also no 'Miss' or 'Mrs'. The appellation for a woman is the same, in other words, if she's married or not. Great, huh?
