PHOENIX—Senate Democrats, against all odds just a few months ago, can see a path that could lead them back to power, with one hitch—the road to a Democratic Senate majority runs through Trump Country.

To win, Democrats must defend 10 Senate seats they hold in states President Donald Trump won in 2016, plus take at least two seats now held by Republicans. In these states, the party faces a tricky balance act. It must appeal to independents and centrist Republicans, who typically decide elections in these states, without alienating...