Depends on the state and the type/definition of private club. In Michigan it's been illegal since 1976 for most family clubs:

>(2) If a private club allows use of its facilities by 1 or more adults per membership, the use must be equally available to all adults entitled to use the facilities under the membership. All classes of membership shall be available without regard to race, color, gender, religion, marital status, or national origin. Memberships that permit use during restricted times may be allowed only if the restricted times apply to all adults using that membership.

Of course, that doesn't prevent discrimination, it just makes it more difficult. We had a friend in the 1980s that was the first Jew at a major country club here. Typically two existing members have to sign the membership request and they simply wouldn't do it for Jews. He was an auto exec, though, and the thought of losing all of the execs from his company changed a few minds.