I've compiled my mod, and I've thrown it on a server I'm hosting, and I'm running into NoSuchMethodExceptions for client @SideOnly methods, and NoClassDefFoundError: net/minecraft/client/Minecraft, which I believe tells me I've created a client-only mod. I don't know how to make it multiplayer-compatible though...how do I do that?

What client-only methods are you using? How to fix it will depend on which ones you're using.

It's an entity that can be controlled like a boat, the method simply gets the user's input and sets booleans accordingly, taken from the EntityBoat class.
 

@SideOnly(Side.CLIENT) public void updateInputs()
    {
    	leftInputDown = GameSettings.isKeyDown(Minecraft.getMinecraft().gameSettings.keyBindLeft);
    	rightInputDown = GameSettings.isKeyDown(Minecraft.getMinecraft().gameSettings.keyBindRight);
    	forwardInputDown = GameSettings.isKeyDown(Minecraft.getMinecraft().gameSettings.keyBindForward);
    	backInputDown = GameSettings.isKeyDown(Minecraft.getMinecraft().gameSettings.keyBindBack);
    }

Strangely enough, I didn't find a call to the EntityBoat's version of the method anywhere within that class, so I added one in mine elsewhere in the code

 public void onUpdate()
    {
    	
    	previousStatus = status;
    	status = getRaceCartStatus();
    	
    	updateInputs();
    	
		...

 which is causing the NoSuchMethodException. If I take out that call, I get the NoClassDefFoundError, looking for net/minecraft/client/Minecraft

For future reference, if you're using Eclipse as your IDE, you can right-click any field or method name and click "Open Call Hierarchy", and Eclipse will show you exactly what classes are referencing the field or calling the method, and where. It's a very useful research and debugging tool