Currently (because the UK is not in Schengen and neither can Eire so long as there is an open border with the UK) Eire maintains an EU/UK border which scrutinises all movements of people entering Eire from anywhere - this is in place of such controls at the Eire NI border and across the Irish Sea. In the event of a hard border at the latter line Eire could, if it wishes to, become a member of Schengen and then the current controls would simply move location. What would be new would have the same consequences all around the EU external border, EU/UK/EU movement of goods and services would come in for scrutiny/controls. As I understand it, in principle Eire would like to join Schengen but is currently in effect prevented in so doing by the UK - on a Brexit without a solution to the Eire/NI border issue, the UK forces the setting up of a hard border which, like all other EU border sections, would be regulated, funded and effectively operated by the EU. Both Eire and the EU have said they don't want a hard border (Eire/NI) and the UK has said they do not either.