A client needs to have Malgun as the font whenever hangul characters are present. I'm trying to find something to use in CSS that is close to it. I was thinking Verdana. Anyone else have a suggestion?
There are no "standard web-based fonts", only fonts that are more or less probable to be installed on the computer, where the browser is running. You may try to build a font-stack, that comes close to the one you want, e.g. the Verdana based font stack from this Sitepoint article, and then use font-loading methods like Google Webfonts to load your defined font for browsers that support loading fonts.
Do not try to give each visitor the same experience, but the best experience possible. Tell your customer, that a website is not a application that looks the same everywhere, but more like a TV program, that must be viewable from a black and white TV also, see this video.