By these measures, Mr. Kaine would seem to stand a real chance of helping Mrs. Clinton.

He’s a statewide official running when there isn’t an incumbent president on the ballot. Virginia is not an especially small state, but it’s not huge either, and the Commonwealth does have a distinctive sense of identity.

One could argue that Mrs. Clinton and Donald Trump are so well known that there’s less room than usual for a vice president to make a difference. In that sense, perhaps it’s more like a race with an incumbent. But one could also argue that Mrs. Clinton and Mr. Trump are both so disliked that a home-state vice president might play a larger role than usual in letting voters feel good about voting for a candidate they might otherwise loathe.

Mr. Kaine is not regarded as a superstar, not even in Virginia. Polls show that his approval ratings are positive, but they’re not well over 50 percent. In 2012, he ran only somewhat ahead of President Obama, although he did have a tough opponent in George Allen, a former senator. He ran further ahead of Mr. Obama in the crucial Richmond suburbs (he was Richmond’s mayor earlier in his career).

Still, if Mr. Kaine could help Mrs. Clinton in Virginia by even a bit, it would be a big deal.

Virginia, which went Republican as recently as 2004, voted almost exactly in line with the nation in 2012 (a 3.87-point win for Mr. Obama vs. a 3.86-point win for him in the national popular vote). Just about everything suggests it could move even further to the left this year. It’s one of the best-educated states in the country — making it especially friendly turf for Mrs. Clinton in a year when her opponent has alienated huge numbers of white voters with college degrees. The state’s unemployment rate is very low, and the D.C. suburbs are home to much of the maligned Washington establishment.

For those same reasons, Mrs. Clinton might not really need Mr. Kaine to secure Virginia. But locking Virginia into the Democratic column with him on the ticket would start making her path to the presidency look easier. If she won Virginia and the states carried by Mr. Obama where she’s not airing advertisements (including Michigan, Maine, Pennsylvania and Wisconsin), she would have 260 electoral votes — just short of the 270 needed to win. She could win with Ohio or North Carolina or Florida. If she lost all of those three states, she could win with any two of Colorado, Nevada, Iowa and New Hampshire.

Mr. Kaine comes with few downsides. He is not regarded as a progressive hero, like Elizabeth Warren. But it is hard to imagine that he will set the ticket back among these voters anymore than Mrs. Clinton’s record already has.

In general, a good rule about vice-presidential candidates is that they don’t matter very much. For that reason, many choose a candidate who will “do no harm.” This time, Mrs. Clinton may have found a running mate who does no harm but takes advantage of the one thing vice presidents probably can do: give a small bump in a key state.