I find it very hard to convince the left of many of our libertarian values, but I’ve had much more success with those further to the right.
For instance, my mother is a more moderate. She still votes Republican, but she’s not opposed to the idea of some government programs for one thing or another. I’ve made great strides in getting her closer to my position simply by talking a lot, and she’s beginning to see a lot of my gripes about regulation now that she and her boyfriend are deer farming.
My father was a Reagan-conservative, but he’s come much closer to libertarianism and I would describe him as a conservative-libertarian. There are only two pieces left of the major libertarian takes that I have yet to convince him of: drugs, and foreign policy. He’s become much more reserved on his foreign policy than he was during the Bush years, and I would say he aligns himself as a bit more hawkish than Rand Paul, but I find that he gets closer to non-interventionism every day. On the drug war, I’ve also gotten him closer to the libertarian position on drugs simply by asking him why if he believes that the government should not be paternal in other facets of life why should they do so on a failed drug war? I think he’s getting much closer.
So I guess the best way to handle it is know who to target and then use their own thinking as a base for your argument.