I am not mainly a historian of economic thought, and I can't speak definitively. But my strong sense is that Murray Rothbard sits on the right-wing fringe of the right-wing fringe, and that his economic thought has had little influence outside of his political corner. He was certainly never a major influence in economics, either academic or in public discourse. Even more mainstream Austrians like Hayek were generally thought of as having rejected mathematical modelling as a method of doing economics, and therefore and placed themselves outside the mainstream entirely. Whether this was justified or not is another question, but post-Samuelson, if you wanted to do econ, you pretty much had to do maths. Rothbard was not just sceptical of that notion, he was viciously hostile to the whole enterprise, and was never shy about making his views known.

Rothbard's output in mainstream research journals is mostly comments rather than full articles. He is discussed positively by a handful of scholars at George Mason (Peter Boettke, Bryan Caplan), but I have never seen a scholar not obviously affiliated with the Mises Institute or that circle more broadly cite him at all, except when writing the history of the movement itself. His political allies are among the fringe (Ron Paul, Pat Buchanan) the odious (Lew Rockwell) and the horrifying (David Duke). Insofar as he has had any influence on government policy, it is indirectly through these channels. But these figures were never more than minority voices in congress and fringe candidates.

In short, his influence has been very limited. There is a group at the Mises Institute who revere him, and an adjacent group of libertarian economists who occasionally engage with his work, but they are themselves well outside the mainstream.