1996 really is too late for Dole. If he had won in 1988 odds are he would have won again in 1992, and overall I'd say he would have been a better president than Bush 41 or Clinton.
In 1980? Well I'd prefer Howard Baker myself, but Dole would have been a solid president (and he likely would have balanced the budget) and perhaps kept the Republicans the small government party they claimed to be. No supply side economics though, as Dole didn't believe in it.
But back to 1996, can Dole get elected? Probably not. Clinton was doing a perfectly respectable job in managing the place, even if he had given up on big ideas, and Dole would have needed something big.
However, we could certainly try. Our POD would be over Gingrich's government shutdown. Quite simply he doesn't tell the "snub" story. Thus Clinton is forced to accept the Republican budget.
This is a body blow to the Clinton approval ratings, which enter freefall for the next three months. Gingrich's high negatives and lack of an organization prevent him running for president (especially against Dole, given their poor relation) but it still propels Dole to the Presidency.
Perhaps somebody else for VP instead of Jack Kemp (could Pete Wilson swing California into the GOP camp for one last election?) or a better campaign—given that his chances are seen as better—or whatever also help.
Senator Dole is elected President, Speaker Gingrich increases the House majority, and perhaps the Senate slides closer to a Republican pick-up.
Presumably a Balanced Budget law is first up (though Dole, to the horror of House Republicans, would be happy to raise taxes to do it).