The following comes from a July 6 Aleteia article by Father Henry Vargas Holguín:
The practice of holding hands while praying the Our Father comes from the Protestant world. The reason is that Protestants do not have the Real Presence of Christ; that is to say, they do not have real and valid sacramental Communion that joins them among themselves and with God. Therefore, they turn to the gesture of holding hands as a moment of communion in community prayer.
During the Mass, we have two important moments: the Consecration and Communion. There – during the Mass – is where we find our unity; that is where we join ourselves to Christ and in Christ, through the common priesthood of the faithful. Holding hands is obviously a distraction from that. We as Catholics are united by receiving Holy Communion, not by holding hands.
There is nothing in the General Instruction of the Roman Missal that indicates that we should hold hands. During the Mass, every gesture is regulated by the Church and its rubrics.
Therefore, we should avoid this practice during the celebration of Mass. Now, if someone wants to do it, let them do it (as an exception) with someone they know very well, without forcing anyone to do it, without making anyone uncomfortable and without trying to convert this practice into a liturgical norm for everyone.
We have to remember that not everyone wants to hold the hand of the person next to them, and trying to impose it on them creates a moment of discomfort to the detriment of prayer, piety and recollection.
This, like other attitudes, is nothing but an exaltation of feelings. Being in communion with someone doesn’t depend so much on holding their hand when praying the Our Father, as on having gone to Confession, being in a state of grace, and, above all, being prepared for the Eucharist.