It is confusing to find that something that was a Good Thing when I was young and naïve has become a Bad Thing. Of course it is two very different things under the same name. I suspect that the same thing has happened regarding the acceptability of the songs we used to sing while drinking beer in college.
Globalization in those days was a mixture of free trade and increasing cultural exchange due to more education, cultural developments like the film industries in small and developing nations and cheaper air fares. The great classical liberal Mario Vargas Llosa celebrated this cultural internationalism and he did not see it as an agent of western domination of the Third World. Education in those days was still by and large educational although it was on the turn. Multiculturalism was happening as a result of postwar migration and the infusion of elite Asian university students in the Columbo Plan. It was not yet weaponised by the left along with the grievance industries.
Even before the internet there was talk of a global village. In the most naïve book ever written some history and policy wonk celebrated the end of history due to the triumph of liberal democracy and democratic capitalism.
The other type of globalization is about world government, driven by the UN and its tentacles with an agenda that has nothing do with liberal democracy.
The point is that libertarians, conservatives and classical liberals need to broadly support the first kind of globalization and fight tooth and nail against the other.