Also speaking to Kuenssberg, shadow home secretary Chris Philp said his party would seek to put in place a "deterrent-style scheme" if it returned to power.
The number of migrants crossing the Channel in Labour's first five months in office is down on the same period in 2022, a record year for arrivals, but similar to the number who crossed in that period in 2021.
Cooper said the number of crossings was still "deeply damaging" and "dangerous", and was undermining border security as well as putting lives at risk.
"Of course we want to continue to progress, of course we want to see the boat crossings come down as rapidly as possible.
"What we are not going to do is deal with this by slogans. Rishi Sunak said he'd stop the boats in a year."
Asked whether the government's reluctance to set public targets on reducing the number of small boat crossings shows it was not a priority, Cooper said: "Quite the opposite.
"We've made clear that border security is actually one of the foundational issues".