How long is an eye infection contagious after starting antibiotic drops?

Depends: If the infection is caused by bacteria, you might see improvement and decreased contagion in a day. If the eye infection is viral, the antibiotic drops won't change the status of contagion at all.

Not very long: Assuming the conjunctivitis is indeed bacterial in origin, it will cease to be contagious in around 2-3 days after starting antibiotic medications.