This isn't really a topic that can be answered true or false.
Technically speaking, everything is some combination of the same elements, so in that respect, everything is connected. However, the Earth is not physically connected to the Sun. It is held in place by the Sun's gravity.
If you're talking about connections to people, then that statement isn't true or false either. The daily activities of a single person in say, Africa, do not affect my life in any way, just as mine don't affect theirs. However, what my neighbors or coworkers do can affect me.
Of course you could consider the "butterfly effect", where the wings of a butterfly fluttering causes a chain of events that lead to a tornado somewhere else in the world. Although, that's just a theory and it hasn't been proven.
Definitely a loaded topic.
"We are all connected; to each other, biologically. To the earth, chemically. To the rest of the universe atomically. We are not figuratively, but literally stardust.” Neil DeGrasse Tyson.
How far can the concept 'everything is connected' be stretched and proven?