Both. As a service provider, it is their responsibility as a business with a service to consumers to make sure that everything is secure. It is no different than car manufacturers making sure their cars are safe to drive.
As a client, it is also your responsibility to make sure that your own interests are secure also, assuming we are talking about computers. Even if you don't use "this providers" service, you should still practice safe strategies to keep everything running smoothly on your end for things you do use. In other words, don't permanently use a PC online without some form of security.
Using the car analogy, just because the manufacturer made sure his product is safe to drive, you must also drive it safely!