But "taking control" of the borders involves border controls. To prevent unwanted immigrants from entering Great Britain from the Republic requires either a hard border within Ireland, or imposing border-like controls on travel between Northern Ireland and Great Britain. The DUP are dead against the latter because it would be seen as a step towards a united Ireland.
One of the many implications of Brexit that its advocates either didn't think of at the time or kept quiet about, and its opponents largely failed to make people aware of.
All I can tell you is that a hard border on the island of Ireland just will not happen despite whatever the rules "should" be.
It is completely impractical given the daily movements of people across the border for work, education and health care, and with no fewer than 300 crossing points, it is impossible to police without blowing roads up again and I don't genuinely think that is going to happen.
This link is to a BBC News video report that tells the story of one cross-border road that crosses the border no less than 4 times in 6 miles - it gives an indication of how difficult this situation is:
http://www.bbc.com/news/av/uk-northern-ireland-40108264/crossing-the-irish-border
Given the significant loss of life during the "Troubles", no politician with any understanding of that be they here in Ireland, or in NI, GB or indeed the EU is going to revert to a situation that could give rise to terrorism raising its head again. That's what a hard border could do.
A political settlement will be reached - politics in Ireland has a long history of compromises being found to deal with seemingly insoluble situations, and frankly I see this as no different. The Good Friday agreement was a prime example of this. Have no doubt, a political fudge will be found that deals with this.
And yes you're right - the Brexiteers didn't give a damn about Northern Ireland or Ireland when they were campaigning on this, in much the same way as most people and politicians in GB don't care about NI in general. It has left both governments with an unholy mess to sort out, which frankly is something the Irish government could do without right now as they have enough issues to deal with.
The Enterprise will continue to operate normally I would predict - there may well be more ID checks at stations for sure, but they already happen on an intelligence led random basis.