Cookman and Sen. Corey Palumbo, chairman of the Judiciary Committee, said they expect to introduce a bill by the end of the week that would move punishments toward home confinement and away from jail sentences.
Sen. Mike Hall, the leader of Senate Republicans, said he likely would support the bill.
"All you're doing is ensuring there will never be child support," Hall said of the existing law. "That's just one of those that just doesn't make any sense at all."
The committee also discussed another slightly unorthodox tweak that they hope could help people find jobs.
Sen. William Laird said West Virginia suspends or revokes the drivers' licenses of 54,000 people each year for not paying fines.
He noted that, in many parts of the state, being unable to drive usually means you can't get a job to pay those fines.
"The bus schedule is pretty skinny in some of the rural areas in West Virginia," Laird said, "and if you don't have a driver's license, you might as well stay home and watch TV."
Laird said revoking licenses for failing to pay small fines is akin to criminalizing a portion of the population and throwing them into poverty.
West Virginia's unemployment rate is 7.4 percent, slightly below the national average. About 25 percent of young people in the state between the ages of 16 and 24 are unemployed and also not in school, according to West Virginia Kids Count.
Laird has proposed a bill providing amnesty to anyone whose license has been suspended for failing to pay a fine that is at least 10 years old. That bill will be considered by the Senate Transportation Committee on Friday.