I’m pretty sure border controls are not under the purview of the Stormont executive, so the opinion of the DUP wouldn’t be particularly relevant. I can’t see strict border controls coming into place if the UK leaves. The CTA (common travel area) that currently exists in the UK, Isle of Man, Channel Islands and Ireland would remain in place, as it is a bilateral agreement between Ireland and the UK and has nothing to do with the EU.
Britain could of course decide to annul the agreement, but I can’t see why they would want to, bad for the economy and bad for the political situation in the North.
Interestingly Ireland has passport checks when arriving from British airports (and some checks via ferry), but the UK does not check travelers from Ireland at all, that could conceivable change, but would only be a minor inconvenience (as they’d just be ensuring one had an Irish or UK passport).
