A private club should have the right to determine who is allowed to become a member.
I agree with you but I also think that this can be quite a tricky issue with some pitfalls if one is not careful.
This reminds me of an incident when I was in college where a white student tried to sue an African American sorority because she was declined admission due to her race. Anyway, I remember reading in the college newspaper that the sorority had the right to decline admission based on race since it was a private organization.
Of course, the use of "private" shouldn't be applied to businesses because it is indeed against the law to decline an applicant for a job based on race or gender. However, notice how there are even exceptions to this rule such as the well publicized case where a guy tried to sue the restaurant chain "Hooters" because he applied for a job as a server and claimed discrimination based on his gender.