"Have you seen Korean movies?"
First, the sentence literally translates to something like "Does the act of watching Korean movies exist?"
The こと means "thing" or "matter" and is used to make everything before it into a noun and accounts for the "act of watching Korean movies" part of the sentence. (The "does" doesn't really exist in the Japanese sentence, but it's required by English grammar.)
The が marks the thing in the sentence performing the action. The action here is existing, which is being done by the phrase that was made a noun, so the が marks the "act of watching Korean movies" as the thing doing the action of existing.
The nuance is a little different. 見たことがありますか means "do you have the experience of seeing ~?" / "have you ever in your life seen ~?", while 見た？ (mita) carries the idea of "did you see ~?", referring to a specific instance > 昨日韓国の映画を見たの？(Kinou kankoku no eiga o mita no?) - Did you see a Korean movie yesterday?
(I don't recommend using か after short forms, because it sounds aggressive and can border on rude.)