Short answer? Probably nothing.
Democrats in Montana now have until Aug. 20 to pick a replacement nominee with former Lt. Gov. John Bohlinger and former state Superintendent of Education Nancy Keenan mentioned. Both would be solid and credible candidates who would almost certainly lose convincingly to Republican Rep. Steve Daines in the fall. That was likely the same fate that awaited Walsh -- pre-plagiarism scandal -- although there was a case to be made (and Democrats made it) that he could have used the power of his short-term incumbency to his advantage. (Some context: Prior to the Walsh decision, Election Lab, the Post's election model, gave Republicans a better than 99 percent chance of winning the race.)
With popular former Gov. Brian Schweitzer (D) out of the running, the picture is much clearer. National Democrats, who weren't heavily financially committed to the race even with Walsh in it, walk away from what looks more and more like an unwinnable contest. Doing so means that Republicans will be in very strong position to net three Democratic seats -- Montana, South Dakota and West Virginia -- and need to win three more seats of a group that includes Alaska, Arkansas, Colorado, Iowa, Louisiana, and North Carolina to gain the majority.
That scenario is the same post-Walsh announcement as it was pre-Walsh announcement. The Montana race is now a slightly more comfortable lock for Republicans. But, the Walsh decision is not a game-changer when it comes to which side will control the Senate next year.