The most such a candidate can do is possibly help win his or her home state. So it makes sense to pick a running mate from a battleground state, which is what Hillary Clinton did when she chose Virginia Senator Tim Kaine.
Hypothetically, if Clinton wins Virginia because of Kaine and then wins in the electoral-college vote by a small margin, it’s possible that Kaine could affect the outcome of the election.
But all these hypothetical possibilities are indeed just that – hypothetical.
Eric Bahrt