The Blades suffered another major blow with a 1-0 defeat by Burnley on Tuesday that moved Sean Dyche's side 15 points clear of their opponents.
United are already 11 points from safety, but Basham said: "We have to have the belief. We might as well throw the towel in now if we didn't have belief.
"I've been at this football club for a long time now and I've been through disappointing times. I've been 14th in League One so it's nice to be playing in the Premier League.
"We're still in the Premier League and we've just got to keep pushing that. We all believe that we should be here.
"Yes, it's going to be really tough and obviously the results and the points tally that we've got shows that it's going to be tough, so we just need to go into every game and be as positive as we can, individually and collectively as a team, and try and get a result.
"As soon as we go off that wagon of being strong and being together, I think that's when you start being worried."
With only two points from 16 games, United have already made the worst start in Premier League history, and they would set another unwanted record if they fail to beat Crystal Palace on Saturday.
No team has ever gone winless through the first 17 games of a season, and Basham thinks the Blades can use that as fuel.
He said: "I think we need as much motivation as we can. Obviously we're all disappointed, we're all feeling down, a little bit of embarrassment there as well.
"Obviously we're very deflated but we want to prove people wrong, we want to try and get the result as soon as possible. We know that teams aren't going to look at Sheffield United and go, 'Oh, they've only got two points, we'll let them have a win'. It's a ruthless league.
"The manager's pushing us all the way to keep that belief there. He believes it's the bounce of a ball, he believes we can keep pushing. We had a lot of 1-0 wins last year so hopefully we can turn it round as soon as possible.
"We don't want Sheffield United to be a club that's recognised for the failures in the Premier League."
Basham is in line to play his 300th game for the club on Saturday, and he said: "I'll always have pride playing for this football club, especially the higher we are in the leagues as well. It's been a great journey for me.
"It's obviously a disappointing time at the moment. Playing for this team and this football club under this manager, it's amazing, and we'll just keep pushing together. I'll take a lot of pride in 300 appearances as well. I'm amazed by that. I've just got to keep going and hopefully I can get a few more."
The 32-year-old defender believes there are positives to the situation and hopes one of his team-mates can turn himself into a hero.
"There's loads to gain and nothing to lose," said Basham. "Hopefully somebody can be the man who wants to take the game by the scruff of the neck.
"We're excited. We're going to try our best to get out of this situation, get as many points as possible and hopefully get some teams looking over their shoulder and saying, 'Sheffield United are sneaking up there'."
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