Hednesford are currently top of the Conference North after eight wins in 11 games, and Barker expects the Pitmen to rise to the occasion as the Staffordshire outfit attempt to reach the second round for the first time in 15 years.
The 38-year-old told the club's official website: "They will be full of confidence because they are top of their division. We had a similar situation with Chelmsford Town last year, so we know that it will be a tough game.
"Without patronising the opposition, this is a cup final to them. They are in the first round proper and they are playing a League One team. Everyone raises it an extra 5%, like we had to against Reading last season.
"There will be half a dozen upsets that day, and we need to make sure that we aren't one of them.
Crawley are sitting in 12th position in League One, and will be aiming to reach the third round of the competition for the fourth year in succession.