Curtis and assistant manager Paul Williams will take charge of Saturday's match against Bournemouth after Bob Bradley's sacking.
Swansea want to appoint their new boss as soon as possible, with ex-Derby manager Paul Clement the first choice.
"We've got to go with somebody who will to get us out of trouble," he said.
"If we were to go down - and there's obviously that possibility - you look at the Championship and it seems to have got a lot stronger. It's not a foregone conclusion [to go straight back up].
"I know Norwich and Burnley have done it but it's difficult. Newcastle look like they'll do it but they're a big, powerful club who can hang on to a lot of their top players. I'm not sure we'd be able to do that.
"If we go down, it's going to be tough to get back."
Swansea were promoted to the Premier League in 2011 and quickly established a reputation as a well-run club with a team that played exciting, possession-based football.
Their promotion capped a remarkable rise from the brink of bankruptcy a decade earlier and almost being relegated out of the Football League altogether in 2003.
Curtis - a former player who was at the club throughout their rise up the divisions - believes the Swans have not been the same since Michael Laudrup left in 2014.
"I think we have [lost our way]. It's just the change of management all the time," Curtis added.
"The best eras were Roberto Martinez, Brendan Rodgers and Michael Laudrup. But we've probably lost a bit of the 'Swansea Way'. It has been diluted.
"We've lost quite a few players too but have we adequately replaced them?"
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Curtis believes the current Swansea squad is good enough to stay up, and the 62-year-old reminds every new signing of how far the club has come.
"I know a lot of the foreign boys have seen [the documentary] Jack to a King. So they've got a brief history of where we were and where we are now," he said.
"They might not be 100% passionate about it but they know where we came from and how important it is [to stay up]. Not just for the club but for the whole area."

The singer-songwriter tweeted on Sunday that he had given his "whole wardrobe [to] a bunch of charity shops" in Suffolk.
Two shops in his home town Framlingham, Suffolk, and one in Ipswich were given several bags each on Wednesday.
Suffolk Family Carers, based in Claydon, near Ipswich, also received about 15 pieces of clothing and plans to sell them via eBay in May.
"It will give fans from around the world a chance to get them," said Anna Haley, from the charity.
She said the charity, which is fundraising with BBC Radio Suffolk for a new mobile carers centre, had been working with Sheeran's mother, Imogen, a jewellery designer.
"His mum has designed a bangle for us, so he's heard about the campaign from her," she said.
Sue Ryder is to sell its items in Framlingham on 18 April.
St Elizabeth Hospice, which also has a shop in the town, said it was still deciding on how its clothing would be sold.
East Anglia's Children's Hospices (EACH), for whom Sheeran is an ambassador, said it planned to hold an online auction, but would be saving some of the items for a sale at its new charity shop, due to open in Framlingham in May.
Swansea's caretaker manager Alan Curtis says there would be no guarantee the club would return to the Premier League if they were relegated this season.
[NEXT_CONCEPT]
A fourth charity has been given a bag of clothes by pop star Ed Sheeran.