There are different kinds of APIs. You seem to be talking about inherently dynamic ones, such as your JSON blob example. Of course you cannot statically guarantee their contents, which is why statically typed JSON layers provide ways of asserting a schema. Using such schema checkers is best practice even in dynamically typed languages.
Either way, you need to handle such failures, both in statically and in dynamically type-checked languages. The only difference is whether the handling happens explicitly throughout your code or automagically by an API that gives you back a well-behaved object. And both of these ways are possible in both static and dynamic languages (but static languages tend to go for the latter, whereas dynamic languages tend to make you do the former).
Either way, you need to handle such failures, both in statically and in dynamically type-checked languages. The only difference is whether the handling happens explicitly throughout your code or automagically by an API that gives you back a well-behaved object. And both of these ways are possible in both static and dynamic languages (but static languages tend to go for the latter, whereas dynamic languages tend to make you do the former).