The borders of most of Europe are open on account of Schengen; you can drive from Budapest to Lisbon without showing a passport or identity card. Britain has decided it wants, not to share in Europe’s common external border, but to become a fortress Island. That is its right.
To date Ireland has fallen in behind Britain’s demands. Yet there is nothing to stop Ireland joining Schengen with one slight modification; i.e. travellers leaving Ireland for other Schengen states would have to go through passport control. This would be to ensure that persons entitled to be in the UK, but not in the Schengen zone, could not leave for other Schengen countries through the Republic of Ireland.
What Britain would do about people entitled to be in the Schengen area but entering the UK over the Northern Ireland border is a problem for the UK, not for Ireland.
NB The advantage of Schengen membership is not just borderless travel within the area, but that only one visa needs to issued to travellers (e.g. from China, Nigeria, etc) who wish to travel in Europe. The country which issues the visa assumes responsibility for the visitor.