Town return to the Broadfield on the back of a narrow away defeat at Wolves and last week's shock win at high-flying Peterborough which moved them up to 11th in League One.
Barker said: "The club are doing their bit to try and increase the gates.
"We have the initiative on Saturday whereby fans of clubs who are not playing because it's international weekend can get a ticket for £10.
"I know a few Brighton season-ticket holders who are coming because they saw how well we played on TV against Wolves and want to come along again.
"If supporters of other clubs want to adopt us as their second team that's great but what we need are the people of Crawley and around the town to come out, hopefully see a good performance and result on Saturday and come back.
"(Peterborough) was a tremendous result which we hope to build on against Gillingham."