Is this a sentence fragment?
With more than sixty percent of the popular vote, incumbent Lyndon Baines Johnson defeating Republican Barry Goldwater, Sr., in the 1964 presidential election.

 - no
 - yes
This is a sentence fragment. It does not express a complete thought.
With more than sixty percent of the popular vote, incumbent Lyndon Baines Johnson defeating Republican Barry Goldwater, Sr., in the 1964 presidential election.
Here is one way to fix the sentence fragment:
With more than sixty percent of the popular vote, incumbent Lyndon Baines Johnson defeated Republican Barry Goldwater, Sr., in the 1964 presidential election.