The Ephremite monasteries, especially Saint Anthony's (which I have visited), began to be attacked by nominal Orthodox Christian parents whose children made the decision to join. These parents were not familiar with Athonite monasticism (or any kind of genuine Orthodox monasticism) and it seemed to them that their kids had joined a cult, like the Moonies or Jonestown. They relabeled many monastic practices, putting them in the worst possible light: to them, monastic obedience was redefined as brain-washing; fasting was food deprivation; midnight prayers was sleep deprivation, etc. These parents were so far removed from traditional Orthodox monasticism that they couldn't recognize it when they saw it.
Some clergy also oppose the Ephremite monasteries because they offer what many parishes have abandoned. When one grows weary of ethnic dancing and ethnic food fairs and the disregard for keeping the fasts, going to confession, and so on that is found in many parishes, the nearby monastery is an appealing alternative but it is an alternative that many find threatening and scary.
Some of these folks have set up anti-Ephremite websites, to which I say:
1. No monastery keeps anyone against their wishes.
2. If American parishes offered more than a watered down version of Orthodoxy they would not be losing people to the nearest Ephremite monastery.*
3. What would you do if your parish priest never heard confessions? Didn't care about the fasts? I know Orthodox Christians who have never been to confession in their entire lives, who, when you bring it up, tell you its a Roman Catholic thing. If this is the state of Orthodoxy in certain sectors of America, then thank God for all genuine monasteries, Ephremite included.
*I don't mean people necessarily become monastics, but come to regard the monastery as the center of their Christian life as lay people.