To begin with, I would like to quote Psalm 1: "Blessed is the man that walketh not in the counsel of the ungodly, nor standeth in the way of sinners, nor sitteth in the seat of the scornful."
One of the clear indicators of "the shameless habits of today's world" is foul language. It is quite easy to encounter a group of teenagers on the street swearing out loud, without any shame. A particular trial for a believing child is to study with peers who behave indecently. These words continue to echo in the child's mind afterwards, tormenting them and robbing them of peace. And what if such behavior is also found in an Orthodox school? I know of a case where a teenager transferred to a regular public school, unable to understand how people who are supposedly believers dare to spew obscene words... Psychologically, it was easier for him to hear such things from unbelievers.
Unfortunately, I also had an experience in my life where a friend preferred the ability to curse over communicating with me. I asked her to either not use those words in my home anymore, or to leave it. She chose the latter. Later, I tried to resolve this conflict and restore our friendship, but there was a blank wall. Thus the Lord delivered me from unnecessary dealings with an unbeliever. It's hard to imitate someone when they are not in your heart or mind. I only wish that my prayer for K. would sometimes come from my soul.
But how does one "observe meekness and propriety always and everywhere" if they have married an unbeliever? More often than not, such believers quickly become secularized. For example, they might prefer going to an event over the Liturgy, in order to please their spouse.
This was already noted back in the 3rd century by the Holy Martyr Cyprian of Carthage, who wrote in his book "On the Fallen" that the reason for the great number of apostates was that they "enter into marriage alliances with unbelievers; they offer the members of Christ to the pagans."