WASHINGTON—Democrat Doug Jones will be sworn in Wednesday as Alabama’s newest senator, bolstering his party’s moderate wing and raising the prospect of growing philosophical divides among Senate Democrats.

After Mr. Jones is seated, Republicans will control 51 seats compared with 49 for Democrats. With that narrow majority, Republicans can lose no more than one GOP vote and still confirm President Donald Trump’s nominees on party-line votes. It also means that Republicans will need Democratic support to pass most legislation,...