It sounds like the 'AF/M' (autofocus/manual) switch on the lens has stopped working. Short of having the lens repaired, there are a few workarounds you can try.

Back Button AF On

In the 60D's "Custom Controls IV" (C.Fn IV -1) menu, [Menu → Custom Functions IV (the fourth line under the orange camera icon tab) → 1. "AF and metering buttons"], change the setting for the shutter button half-press from the default 'Metering & AF Start' to 'Metering Start'. This means that half-pressing the shutter button will NOT initiate autofocus. Ever. Under any circumstances.

You'll need to assign another button to initiate and maintain autofocus when you desire to use AF. By default, the 60D is set to use the 'AF-ON' button near the upper right corner of the back of the camera to initiate AF. You can also assign the AE-L (Auto-Exposure Lock) button marked with a star '*' symbol to provide the "AF-ON" function.

Once you've set the camera up this way, you'll need to press the back button you have selected for "Metering and AF start" anytime you want the camera to AF. If you release the button, the camera will stop performing AF and leave the focus position of the lens where it was at when you released the button unless you move the focusing ring.

• If you've got the AF mode set to "One Shot AF" once AF is locked there won't be any difference between continuing to hold the back button or releasing it until you've fully pressed the shutter button to take a picture. (The "AF confirmation" dot in the lower right corner of the viewfinder will stop showing if you release the button after AF Lock has been achieved.)
• If you've got the AF mode set to "AI Servo AF" then Servo AF will be active when the back button you've set to "AF-ON" is pressed and not active when the button is not pressed. When the back button is released the lens will stay wherever it was focused when you stopped pressing the button unless you manually turn the focusing ring.

I prefer to swap the functions of the 'AF-ON' and 'AE-L' buttons on all of my Canon bodies. I have the 'AF-ON' button set to do "AE Lock (Hold)" and the 'AE-L' button set to "Metering and AF start". I do this because my thumb more naturally falls above the 'AE-L button' than the 'AF-ON' button and I use back button AF for almost every frame I take, while only occasionally using the AF-L function since I normally use Manual exposure mode. YMMV.

Back Button AF Off

The other most obvious option is to leave the shutter button half press set to "Metering and AF start" and set the 'AF-ON' or 'AE-L' button to "AF-OFF". Any time you press the back button you've set to "AF-OFF" and hold it down AF will be paused and allow you to manually focus your lens. You'll need to apply continuous pressure to the button set to "AF-OFF" to maintain the manually selected focus position. If you pull your thumb off the back button set to "AF-Off" while holding the shutter button at half-press, the camera will again resume AF while the shutter button half-press is maintained.