Oppose intrusions into privacy as vociferously as you would if it had been proposed by the Clinton administration. There were several useful reforms in the USA Patriot Act enacted along with some threatening expansions of the government’s surveillance powers over the internet that were completely unrelated to terrorism. Republicans lose credibility as defenders of limited government when they go mute on this issue.
Oppose intrusions upon the Bill of Rights more consistently in Congress. Democrats used to have a far bigger edge over Republicans on the issue of free speech than they do now that their left has endorsed restrictions on politically incorrect speech on campuses and elsewhere and pushed for campaign regulations criminalizing political speech. Still, Republicans in Congress should be more principled supporters of the First Amendment than they sometimes are. And this goes as well for the Second Amendment where Republicans in the past have relented to demands for various unreasonable regulations on gun ownership or use, and asset forfeiture laws that some but not enough Republicans have opposed. When libertarians do not trust Republican legislators to respect the Bill of Rights, they will be more likely to vote Libertarian.
Nominate more libertarian-conservative judges like Clarence Thomas to the courts who care about protecting individual liberty, not just traditionalist-conservative judges like Robert Bork who care most about the “liberty” of the majority to enshrine its preferences into law. (His words not mine.) Appoint judges who care about federalism, the Second Amendment, and also about the First Amendment and the unenumerated rights “retained by the people” referred to in the Ninth Amendment. The more Republicans do this, the more trusted they will be by libertarians and I cannot imagine this costing them votes from the middle swing voters. The more they fail to do this, the more votes they will lose to Libertarian-party candidates.
Care about federalism in the Congress. Republican politicians are well known for raising federalism concerns only when they oppose a particular legislation on other grounds. Try being a little more consistent. Refrain from proposing legislation that exceeds the enumerated powers of Congress in Article I of the Constitution. Though at times they may risk losing swing voters on this issue, the more legislation they support that exceeds the powers delegated to Congress by the Constitution, the more votes they will lose to Libertarian-party candidates.
Care more about the free market. Republicans pay the political price for being pro-business. Unfortunately, this often does not mean their policies are pro-free market. The Farm Bill comes to mind. When Republicans refuse to cut corporate welfare, they not only act in ways that reduces their appeal to moderates, they lose votes to the Libertarian party.
Stop making snide gratuitous remarks about libertarians. Nothing turns off libertarians more than the sort of wholly gratuitous snide remarks about libertarians in conservative publications. By gratuitous I mean they show up even in articles about policies with which libertarians and conservatives agree. The more libertarians feel unwelcome in the coalition that is the Republican party, the more they will vote Libertarian.
Back off Prohibition. OK, I realize this like abortion is a sensitive subject — or should be — and one that has serious political repercussions. The Republican coalition is, after all, a coalition and libertarians if they are inside the tent cannot be expected to call all the shots. But drug prohibition should be an issue about which reasonable Republicans may disagree — like abortion — or libertarians who care about the serious social and personal costs of Prohibition will be driven outside the tent. At minimum, Republicans should support letting states decide this question of crime and punishment when it concerns the wholly intrastate commerce in drugs whether for medical or recreational purposes. Supporting rather than opposing state choice on this issue would make the Republican party far more libertarian than the drug-war-mongering Democrats without having to support legalization. If Republicans added this to their federalism concerns, they would attract rather than lose votes to the Libertarian party.
Unless, of course, Republicans don’t really want those extra senators and governors.