While multiplayer mode can ruin other genres, I think in strategy a multiplayer mode usually makes games better. At least if multiplayer is in it's core.
The reason for that is that it needs better balancing and game mechanics to work.
And I think Civilization is actually a great example. Even though it has a multiplayer mode, it is not a multiplayer game in it's core. If it was, it would work better.
But because the game mechanics are now either catering directly to the player, the game will just become "un-fun" if you are looking for a challenging experience.
The best example is Civ 5 and it's Culture and Diplomacy Vicotry, which the AI cannot win and which will break multiplayer games (if you know what you are doing). So it feels like using a cheat in Singleplayer and you need houserules to avoid it in multiplayer. If this game was designed with multiplayer in mind right away, something like that should not have happened and would become more fun.