Holding hands during the Our Father seems to have emerged from Alcoholics Anonymous meetings, where it remains a common practice today. That’s why it was associated with the Our Father rather than the Creed, or intercessory prayers, etc. While it is not forbidden in the norms for liturgy, it is inappropriate for a couple of reasons.
In the first place, it is imposing on other people who may be uncomfortable with the practice. To some, holding hands signifies an intimacy that they do not want to share with strangers. Others worry about germs on the hands of others. So there is a kind of boundary violation when whole communities hold hands during the Our Father. It is one thing for spouses or families to hold hands on their own initiative, but communal custom and insistence that everyone join hands creates a pressure to do so that is inappropriate.
It is also problematic insofar as it introduces a practice in the Mass that is not indicated in the norms. As such it is disruptive to the liturgy. In effect it becomes a rather acrobatic gesture that involves reaching across aisles and backward toward people in the pew behind, etc.
Again, holding hands during the Our Father is not forbidden, since the norms are silent about the people’s posture at the Our Father, other than that they stand. The priest alone is required to extend his hands during the prayer. But silence does not mean that anything is allowable, and given that some people find this intrusive and that it does tend to introduce a disruption in the liturgy, it seems an unwise practice and surely something that no priest should actively instruct the congregation to do.
(Source: Our Sunday Visitor)
Read more in the “Why Do We Do That?” series from Deacon Mike Fritz.