The trend of the discussion seems to be towards this helping Romney. I'll propose the opposite possibility: that it will end up hurting Romney. Either possibility may be valid, but I'll go with that idea, and I believe there is a very strong argument for it.
Romney will be running with a ticket destined to lose, and destined to lose hard. Goldwater will be smeared based on things he said or extrapolations of things he's said and done exploited for campaign commercials and campaign literature, just as he was in the actual election. Romney has a very strong possibility of also being stained by running with Goldwater in such a situation. Although you could make the argument that Romney would be able to survive attacks on Goldwater, or that Romney would give a legitimacy to stave off those attacks. At the same time, Johnson could also direct attacks against Romney that would scar him for the remainder of his political life.
Romney will also have a strong prospect to lose legitimacy with and of being of the Rockefeller wing, in part at least, by running with the figure and faction the Rockefeller wing rebuked and perhaps could be said to revile. And those Conservatives in turn booed Rockefeller and molested Ike's niece on the convention floor. The scathing attacks on Goldwater in the election of 1964 came not just from Johnson, but came from the Moderate to Liberal Republicans and leading figures like Rockefeller and Scranton. LBJ didn't invent things out of the blue; he utilized an existing framework of dissatisfaction and a preexisting list of grievances, along with things Goldwater was already known for and thought of as.
Unless he can sell it as a ticket of unity instead of one where he's going along with the Rockefeller's anti-Christ, there is the strong prospect for it to be damaging. Taking into account the vitriolic ideological conflict that was 1964 for the Republican party, I don't know if he could. In OTL 1968, Romney had the benefit (initially) of having Rockefeller's blessing and backing, and being like Rockefeller-Lite. He'd likely not have that blessing in this 1968, possibly suffer from Rockefeller getting into the race from the get go, and would have legitimacy issues with the Rockefeller label.
There's been mention of 1964 being a chance for Romney to work out the kinks and get better acquainted with national media and stop being such a gaffer, therefore going into 1968 strong. I don't know how much water that holds. There is a very strong chance for Romney to gaffe on the campaign trail. 1964 is too early for the brainwashing gaffe, given he hasn't yet been by the generals, nor is Vietnam really an American war to be brainwashed about, let alone be rejected after years of soul searching. However, there is a strong likelihood for something. Getting out there may not strengthen him politically. It may make the public and/or the party view him as a problematic gaffer and a political paper tiger: someone who looks perfect on paper (the only man Jack Kennedy worried about running against in 1964), but in practice fails, or at least is perceived to fail.
And in 1964, Goldwater is going to lose and will do so by a crippling number. That does not seem to set up Romney well for 1968. He could be considered the initial front runner, but it seems like he'd have the prospect of strongly losing ground as things became less nebulous and you had a field of candidates come into play.