The Constitution does not have anything to do with private actors!!!!! State owned property is governed by the Constitution and therefore Constitutional rights apply there, but they don't (at least shouldn't) apply to private property, whether or not it is "open access". If I own a piece of property, I can decided who I let on it and under what circumstances. If I want to say no blue shirts, no firearms, no groups of teenagers, no preaching communism, that is fine. Nobody has any right to do any of these things on somebody else's private property. See, the thing is, when government says "no preaching communism", if you disobey you will have force used against you (thrown in jail, fined, or whatnot). If you break my private property rule, all I can do is bar you from my property, I can't initiate force against you. If you persist to the point where you've initiated force against me, I can have the government charge you with trespassing, which is then force in response.
Trying to claim that businesses with "open access" have to respect the Bill of Rights is a terribly dangerous idea. If a newspaper that prints reader's comments refuses to print a communist rant, can the commie file suit claiming a 1st amendment violation? If a radio station only puts conservatives on the air, can a liberal sue claiming his free speech was censored? What about a club or bar with a dress code, is that a 1st amendment violation? Saying the Bill of Rights applies to any kind of private property is an egregious violation of property rights; the right to property is necessary for the right to life and is the core of all other rights. Destroying property rights destroys all rights, even the right to life. That's why protecting property rights is of utmost importance and why liberty and socialism cannot coexist.
Also, "where would you draw the line? do you feel that we should go back to 'no blacks' or 'Irish need not apply'?" has such a biased phrasing that I'm not going to give it a proper answer, but you ought to be able to figure out my response.