The gold standard in security is: Validate your inputs, and encode, not sanitize, your outputs.
First, validate the input server side. A good example of this would be a phone number field on a user profile. Phone numbers should only consist of digits, dashes, and perhaps a +. So why allow users to submit letters, special characters, etc? It only increases attack surface. So validate that field as strictly as you can, and reject bad inputs.
Second, encode the output according to its output context. I'd recommend doing this step server side as well, but it's relatively safe to do client side as long as you are using a good, well tested front-end framework. The main problem with sanitization is that different contexts have different requirements for safety. To prevent XSS when you are injecting user data into an HTML attribute, directly into the page, or into a script tag, you need to do different things. So, you create specific output encoders based on the output context. Entity encode for the HTML context, JSON.stringify for the script context, etc.