Here's something from the BBC live feed...
Her statement is misleading on a number of levels.
1) The problem is not solely the common travel area between the UK and Ireland, it's also the freedom of movement area between Ireland and all other EU states (since anyone wanting to get into the UK can go via Ireland->N.I.->Rest of UK without being subject to the same level of scrutiny as they would face coming through a port/airport in mainland UK) The two issues go hand in hand, and can't be separated. Unless Ireland gives up its own freedom of movement, the flow I just described will remain open to people.
2) The old free movement areas pre-EU were much, much smaller than they are now, encompassing fewer countries, so for example there was no route such as the ones the refugees are taking now that passed solely through "free movement" countries back in those days. In other words, the world has changed too much in the interim for her throw-back to the past to be meaningful as a comparison.
If you're in favour of Leave, how would you see the Ireland/Northern Ireland border problem resolved? Do you feel that re-establishing border posts there is a worthwhile price to pay?
(BTW, what I posted above about a possible route for people to get into the UK is factual based on the current situation, not simply my opinion. There's no border check between Ireland and Northern Ireland. There's also no requirement to have a passport for travel between Northern Ireland and mainland UK, and a quick Google shows that no more than occasional spot checks of photo ID are conducted for ferries arriving from N.I. Please do your own research if you have any doubts about this - everything can be confirmed in under 5 minutes.)