Question

The base or root of a phylogenetic tree is where a population either becomes extinct or is currently living today

Answer 1

@markbankhead Neither. Is this a question from school? It is also impossible to answer your question without more info.

Answer 2

Here are some of the issues:

the root and the base are not the same thing in phylogenetic theory.
populations are not on trees, there are only species trees and gene trees
the nature of a phylogenetic tree means that there can not be any extinct species used to make the tree