Not really a problem often. Cats with light fur tend to have white nails and you can see the quick (the living part of the nail) really easily. There are also methods on how to cut the nail properly, with correct angles and specialized scissors. Even with dark-coloured claws, it isn't hard to avoid it. Simply trimming the extremely sharp hook end is enough, you don't need to cut half their claw off.

Declawing means removing a digit in a cats' paw. It is a completely different process to what I'm describing here and is a horrific thing to do to a cat, basically maiming it in the process. It should only be done as last resort in illness to save a cats life, not to prevent it from scratching. It is basically the difference between cutting your nails so they are short, and cutting the end of your fingers so there is no nail to cut.