Short answer: Yes
Longer answer:
Yes, but that shouldn't be interpreted as a suggestion to start by writing your classes with everything private; that approach implies class design by focusing on the implementation detail before you've settled on an interface.
One of the most important aspects to consider when designing a class is how it will be used; which involves thinking about your public methods before you start thinking about private/implementation details.
Furthermore, that approach is usually missing out on chances to ask yourself "How would I write a unit test for this class?" - which is an important question to ask even if you aren't actually writing unit tests. (Related: "What are the design principles that promote testable code?""What are the design principles that promote testable code?" )
So, once you have defined the public interface, then it is a good idea to default the rest to private because most of that will typically be gritty implementation detail which is of no concern to anything outside of the class.