No. A dirty cabin air filter may restrict a tiny bit of air flow and may cause debris in your cabin from dust and dirt and pollen. But it will not affect cooling.
RepairPal Certified shops can diagnose and repair your 2002 Ford Explorer Sport Trac. They provide transparent pricing with a minimum 12-month or 12,000-mile warranty on repairs and services.
Does the 2002 Sports Trac have a cabin filter and if so where is it located?