Town Council members said they are concerned the trash can become a refuge for rats.
Hilton also said that the sight of trash sitting out does not present a "welcoming sight" for visitors.
Charles Town Police Chief Mike Aldridge said officers are investigating one case involving trash piling up in town.
Aldridge said he would like to see a crackdown on the trash problem, with the municipal judge imposing heavy fines as a deterrent against future violations.
"It may hurt a few people in the process but people will see it," he said.
Some of the garbage offenses carry up to a maximum fine of $1,000 and 30 days in jail, according to the town ordinance.
City Manager Jane Arnett said the town receives about six to 12 calls a week from residents complaining about trash piled up on their neighbors' property.
She said many of the calls involve the same neighbors continuously.
The town has a contract with Waste Management to pick up the garbage. The garbage firm bills the customers, but does not pick up garbage of those who have not paid their collection fees.
Councilman Randy Breeden said there is a town ordinance that requires residents to pay their garbage collection fees or be fined from $100 to $500.