It's not just relations with Russia: In a world where power has dispersed and fractured, a president who's perceived as weak, whose deals are seen as unreliable, is not good for the United States.
If the New START treaty really did damage U.S. security, none of this would matter. Treaties shouldn't be ratified for their own sake. But there is nothing harmful in this treaty. Nobody inside the Senate has any problems with the treaty itself. The Republican whip, Sen. Jon Kyl of Arizona, who has emerged as the crucial player (the GOP caucus will follow his lead on ratification, one way or the other), is using the treaty as a bargaining chip to prod Obama to spend more money on missile defense and "nuclear modernization" (i.e., maintaining the arsenal, including the replacement of aging ICBMs if necessary). Obama may have to go along with this to get the votes—though the question is whether Kyl will vote to ratify if he gets what he's asking for or whether he's just playing budget games. Neither the White House nor the Senate's Democratic leaders know the answer.
That's the real debate that's going on—though it might be good to get it out in the open, so the deficit hawks can decide whether they want to spend tens of billions of dollars, beyond the tens of billions that Obama is already spending, to shore up a nuclear arsenal that's already far more powerful than national security requires.
To most Republican opponents, the treaty itself is irrelevant—though the Senate's failure to ratify it would do more real damage to U.S. security than even the most twisted reading of the treaty's language ever could.