どうせ is a way of saying 'anyway', 'in any case', or 'at any rate', but with a nuance of resignation or pessimism. The speaker is saying that regardless of efforts or alternatives, the result is the same, often something not good. It's common in conversation when you want to justify not trying hard or to point out a harsh truth. For example, you might say 'どうせできない' (I can't do it anyway) to give up before trying. It sets a tone that the following statement is something you've accepted, even if reluctantly. It's often used to brush off hope or to show that you're not naive about a situation. In writing, it appears in essays or narratives to convey a character's cynical or realistic viewpoint.