I'll try to do both sides justice although I'm sure there are people representing either side that could correct me on many aspects...
I think of 'globalist' as being something of a perjorative/having at least a slight negative connotation. It describes a perspective where all humans are part of a global community/family. You could contrast it with 'nationalism' (which has it's own negative connotations--the two sides provide derision against each other) in which nations should be independent and sovereign.
Critics of globalism argue that it centralizes too much control, planning, finances and government, leaving locals with less autonomy. Nationalists are not fans of e.g., the United Nations, the World Trade Organization, etc.
Critics of nationalism argue that international cooperation brings many benefits and that nationalists are merely racist or xenophobic. It's harder to list organizations that globalists are not a fan of because most nations have at least one nationalist movement. Brexit would be a good example in the Anglosphere.