Not a lot of business majors in this bunch, I see.
Courses fit into three groups for relevant purposes: Private clubs (which in turn could be broken down, but not really relevant to the question), privately-owned public courses (or revenue-oriented municipal courses), "draw" courses, subsidized public courses.
The barrier for the first group is obvious. The second group is generally trying to maximize revenue. The owner isn't raising the price to keep people off, you know.
And for the "draw" courses, I mean courses that subsidize golf for those who come to play in order to profit off a related good - usually a hotel (but sometimes real estate). These create barriers by limiting access to those who purchase the bundled good.
Amongst public courses, only the last might consider creating a non-price barrier - and even then, would only do so to the extent that it would benefit its constitutents. Even then, they are unlikely to do anything to dissuade constituents to go. For those purposes, municipal courses also do the obvious - they price discriminate against non-constitutents (or even restrict tee times entirely).
What is interestingly posed by JK's post is the idea that a golf course could be a Giffin Good. If the price is so low that the course becomes too crowded, demand will actually decrease since nobody will want to play the course given the time restraints. Only when the course is raised to a price which naturally filters out some people anyway would (in theory) the golfing public want to play. However, outside of those courses for which playing is simply a display of wealth, I do not think the pace of play issues actually result in any courses being Giffen Goods.
Anything else may lead to some courses that are unfortunately less crowded than the operator might want - but trust me, the operator isn't then thinking of ways to keep it like that!