Dempster will swap Motherwell for Leith this summer after she was recruited by Hibs chairman Rod Petrie to revitalise the Edinburgh outfit.
A miserable season for the capital club's supporters has seen crowds plummet while a run of 10 games without a win has put Butcher's side in danger of a Scottish Premiership relegation play-off.
With just a single-point cushion over Kilmarnock, their remaining fixtures against Partick, Ross County and the Rugby Park men will be crucial.
Dempster's five-year spell in charge of the Steelmen could end with a third consecutive top-three finish but Butcher knows it is up to him to solve Hibs' real problems.
He said: "I think everything is sorted out pretty well off the park - there are not many areas that we need to develop.
"The major thing we need to fix is on the pitch."
Dempster will begin work on June 1 but her first task will be to build a relationship with the former Well boss.
Butcher said: "I don't know Leeann, she joined Motherwell after I left. But she has done a terrific job by all accounts, so she will be a welcome addition to the Hibs administration set-up.
"I saw she has been labelled the club's first summer signing and I'm looking forward to working with her.
"My relationship with the chairman has been excellent and with Leeann coming on board that will take a bit of the burden off him.
"But that is all in the future. I'm looking forward to working with her but that will be next year. We still have three games to play.
"While it is nice to have Leeann coming in, right now is all that matters."
Despite losing to city rivals Hearts for a fourth time this season last Sunday, Butcher believes his side are closing in on that long-awaited win.
They face Thistle on Saturday at Easter Road and Butcher said: "I feel as if there is light at the end of the tunnel. It's the first time for a long while I've felt like that.
"Normally if you lose a derby match it is all doom and gloom but I didn't see it that way.
"The way we played gives us great hope and optimism starting for the final three games.
"There were lots of bright spells against Hearts, the second half in particular when it was just wave after wave of attack. How we didn't get anything from the game I don't know.
"But at least we got a goal and we certainly got stuck in. The foul count was two to one in our favour, which is what our crowd wants.
"The way things have been going, any chink of light we have to grab hold of and nurture. In the sea of despondency a lot of people say we are in - but I don't - it certainly gives us optimism."