Keep a static collection of red routes and use that in your check. For example

// route-styles.js
export const RED_ROUTES = new Set([
  "/browse",
  "/some-other-route",
  // etc
]);

// Header.jsx
import { useRouter } from "next/router";
import { RED_ROUTES } from "./route-styles";

function Header() {
  const router = useRouter();
  return (
    <ul>
      <li
        style={{
          color : RED_ROUTES.has(router.route) ? "red" : "black" 
        }}
      >Resut</li>
    </ul>
  );
}

Another option might be to create a map of routes to colour. That would let you use more than just the two choices

// route-styles.js
export const COLORS = {
  "/browse": "red",
  "/some-other-route": "red",
  "/the-blue-one": "blue"
};

and in your component...

<li
  style={{
    color: COLORS[router.route] ?? "black"
  }}
>

Maybe a array would work?

import { useRouter } from "next/router";
import styled from "styled-components";

function Header() {
  const router = useRouter();
  const theRoutes = ["home", "browser", "view", "about"]
  return (
    <>
      <ul>
        {theRoutes.map(route =>
          <li style={{ color: router.route === route ? "red" : "black" key={route}>{route}</li>
        )}
      </ul>
    </>
  )
}

You could pass a prop to Header from page

export function Page () {
   return (
         <MainPage>
            // if you have BaseLayout you have to pass it to Baselayout and from there to Header
            // addiing only `red` prop will be evaluated red=True
            <Header red />

         <MainPage/>
   )
}

In your Header

     <li style={{ color : props.red ? "red" : "black" }}>Resut</li>