So the communist party only accepts the most advanced elements of the proletariat, people with the highest class consciousness. How does the party educate the rest of the population if it is that picky with who they let in?
I was told that the difference between the Bolsheviks and the Menscheviks was that the Menscheviks let anyone join, but the Bolsheviks only let people with advanced class consciousness join. How does that give popularity to a party if it is so exclusive?
The popularity of the party is not dependent on it having large ranks, but from its effective party work in critical sectors.
Lenin and the Bolsheviks argued for a professionalised vanguard party of dedicated individuals, so not only do they have advanced class consciousness but they dedicate their lives to bringing about revolution. So support for the party from outside should come from the effectiveness of this strategy, though it's crucial to note that the active agent here is the party for the workers and the support will materialise itself simply as a natural reaction to correct and effective party actions. This is true for communist parties both inside and outside power.
On the other hand the Mensheviks (and later the Trotskyists) argued for mass participation, which is also common in social-democractic parties today. This dillutes the quality of the party work, as militants will not be normally expected the level of commitment from a vanguard party, which among other things leads to a less effective party (and therefore less interest from outside). Other than that, another key point is that it also harms party discipline and theoretical consistency, leading to all sorts of sectarian and revisionist infighting that the principle democratic centralism is meant to prevent.