Ireland doesn't have immigration problems.
The issue with the Irish border is both simple and complex. The simple version is 100 years ago, Ireland was fighting for independence from British Rule. The majority of (Catholic) Ireland wanted independence and a united Ireland. The majority of (Protestant) Ireland - located in the northern provinces wanted to remain under British rule.
The deal enforced by the British was the partition of Northern Ireland, which to this day forms part of the United Kingdom of Great Britain and Ireland. The southern part of Ireland, gained independence in 1922 and is known as the Republic of Ireland, or Éire. It is a completely independent country from the UK in the same way that France, or China, or Bolivia is.
However, since then there has been significant unrest in Ireland. You will have heard of the IRA, I'm sure. The IRA were/are a terrorist unit who were campaigning for the British to get out of Northern Ireland and for Ireland to be united again. The Northern Irish side also had terrorist units, fighting against the IRA.
The situation was not helped by the hard border between Northern Ireland and the Republic. It is only since 1998, with the signing of the Good Friday Agreement, that there has been anything resembling peace in Northern Ireland and the Republic. At that time, the border was abolished and there has been freedom of movement between both countries. Neither side wishes to violate the Agreement, but tensions are high and the political atmosphere is unstable.
Crucially, the GFA states that there will be no hard border between Northern Ireland and the Republic. However, if Brexit goes ahead, Northern Ireland will leave the EU at the same time as the UK. This exposes the Republic of Ireland as they will remain in the EU, with an unprotected border. If either Northern or Southern Ireland implement a border, the Good Friday Agreement will be nullified and we will see a return to terrorism and loss of (innocent) life.
The Donald's quote demonstrates he has no clue about any of it - even on a superficial level. Read it aloud. If you heard someone else say those words, in that order, would you be filled with confidence that it will all work out okay?
Sure, I get the point that he's not interested in the Irish border. It's not his problem. But he is supposed to be aware of politically sensitive issues and foreign matters and he's obviously out of his depth. Now, I have two thoughts on this:
1. He should have had an answer prepared as it was quite likely he'd be asked about it - his answer makes him sound clueless - it means nothing at all... bigly.
2. Why are his staff letting him be interviewed without being sufficiently briefed?
And, Doc - this was not an attack on YOU (and by the way, over here, we consider it quite rude WHEN PEOPLE USE CAPS).