A border is the edge of something. In your example, "borders" means the edges of countries. So doctors without borders treat patients in any country, without paying attention to country lines. It's similar with reporters without borders.

According to Webster's New World thesaurus, synonyms are: hem, end, trim, fringe, frontier, boundary, outpost, perimeter. In my opinion though, 'outpost' and 'frontier' have somewhat different meanings.