Anthem customers who enrolled through the Connecticut health insurance exchange between Oct. 1 and Dec. 23 are still having trouble confirming that the company received their first monthly premium payment. Others customers are struggling to get an ID card, but things are getting better, according to insurance regulators.

Earlier this week, Anthem extended its payment deadline a second time to Jan. 31. It also set up a Connecticut-specific customer service email Wednesday for questions related to application and payment status, prescription benefits, and provider participation. No member ID number is needed in order to send an email.

Deputy Insurance Commissioner Anne Melissa Dowling said she thinks the company reached a “turning point” because the call times they’re experiencing are beginning to go down.

She said a week ago there was concern because the vast majority of Anthem customers who had paid their bills still didn’t have proof of enrollment. Dowling said the company has been in touch with the Insurance Department daily, giving them updates.

Anthem’s parent company, WellPoint Inc., had difficulty enrolling people in 14 states and its customer service line was overwhelmed with phone calls.

Sarah Yeager, an Anthem spokeswoman, said that call times are coming down and we are working hard to reduce them further. There are “1,000 associates staffing our call centers to respond to ACA-related coverage questions and concerns,” she said.

“We understand that members have questions and some are experiencing delays in receiving invoices and ID cards, as well as reaching our customer service team,” Yeager said. “Our goal is to ensure our members can access their benefits as early as possible in 2014.”

Dowling said it’s unclear how many Connecticut consumers have been unable to get medical services or access to prescriptions as a result of the problems Anthem has faced. Anthem is the largest of the three insurers offering plans on Connecticut’s exchange and they account for about two-thirds of the 34,295 individuals who enrolled before Dec. 23.

The call volume to the Insurance Department’s consumer affairs division has begun to drop, but for the past few weeks — as the first Jan. 15 payment deadline approached — they were getting 50 calls per day related solely to issues with Anthem.

“They, clearly on a national basis, were underprepared. That is a disappointment,” Gov. Dannel P. Malloy, said Wednesday at an event in South Windsor. “They, like the federal government, decided to have a single system, in this case to represent all 14 markets that they were active in. I think the surge that they saw particularly in December, they were underprepared for. That’s a reality.”

Malloy met with the company’s president of Connecticut operations in his office Tuesday with Lt. Gov. Nancy Wyman and Kevin Counihan, the CEO of Access Health CT. He said he was confident the insurer was making “substantial progress.”

“As of yesterday, 45 percent of payments for people who have signed up have actually been processed and that number is expected to rise rapidly,” he said.

Malloy said he suggested that Anthem should reach out to the public and explain that the deadline to make the first premium payment has been extended to Jan. 31.

“I have urged them to go on TV and in print ads to explain that situation and what people can do to resolve their worry because people want to know that they’re covered,” he said.

Malloy said the insurer also has opened up a walk-in center where customers can make a payment or receive their insurance card.

The governor also suggested that the incident presents an opportunity for new Anthem jobs in Connecticut.

“I am happy that the call center that serves this part of the country is actually based in Wallingford, Connecticut and those jobs are in Connecticut and clearly they’re going to have to add more jobs in Connecticut,” he said. “So on that side I’m pretty happy.”

Hugh McQuaid contributed to this report.