Air Carrier Captain reported a fume event during departure climb. Flight Attendant notified flight deck of a 'strong musty odor' in the aft cabin. Flight crew executed the appropriate QRH procedures and requested priority handling for a diversion.

Narrative

After a normal taxi out and departure from ZZZ we were given a climb to 10;000 feet by ZZZ departure control. ZZZ then handed us off to Center and we were given a climb to FL240. Shortly after beginning the climb out of 10;000 feet we received a call from the #1 flight attendant stating they were getting a strong; musty odor in the aft section of the cabin. I told the flight attendant we would begin our procedures for fumes and asked them to let me know if there is any worsening or improvement of the smell. I ended the call and told the FO (First Officer) who was also the PF (Pilot Flying) what the situation was. I assigned PF and radio duties to the FO and began the immediate action items on the QRC. I also instructed the FO to request a level off and prepare ATC for a diversion. After following the QRH guidance I contacted the flight attendants to get an update on the fumes in the cabin. I was told that the fumes had not subsided and could be smelled throughout the entire cabin. I was also informed that portable oxygen bottles were being used by flight attendants in the rear of the aircraft. At this point I instructed the FO to [request priority]; and let ATC know we would likely be diverting into ZZZ1. Then I contacted our Dispatcher via the crew phone app and quickly summarized our situation. I asked for assistance in determining the best diversion airport and we quickly agreed on ZZZ1 based on the current weather and our proximity at that time. After contacting dispatch I informed the flight attendants that we are diverting to ZZZ1 and would be on the ground in 15-20 mins. I asked if the fumes had diminished and was told there was no difference; and some of the flight attendants had temporarily moved toward the forward cabin area to escape the strongest odor. At this point I elected to complete the Smoke/Fumes removal procedure. After completing this process I again asked the flight attendants about the level of fumes in the cabin and they stated that they were diminishing. Even though the fumes were beginning to subside; I still believed the safest course of action was to land; even though we would be overweight. During the time frame when I was consumed with QRH procedures and communications with dispatch and the flight attendants; the FO was managing the aircraft's flight path and loading the approach procedure for runway XXL at ZZZ1. We completed the Non-Routine landing and overweight landing checklists and a landing performance assessment. The first officer briefed the approach and I elected to have him fly the approach and landing. I asked ATC to advise ARFF (Air Rescue Firefighting) of our overweight landing so we could have a brake and tire inspection after arrival. The First Officer flew a perfect approach with a very soft touchdown at a vertical speed of less than 100 FPM. After rollout and deceleration we transferred control; and I was able to taxi clear of the runway. ARFF made a brief inspection of our wheels and brakes and did not note any damage of excessive temperatures. We taxied to the gate and prepared to deplane the passengers. At the gate our flight was boarded by members of ARFF; and they would not let anyone leave the aircraft until they collected air samples. Once the air samples were collected and determined to be safe; they allowed all passengers to deplane. When the aircraft was completely deplaned the flight attendants informed me that they were not feeling well and wanted medical attention. They were transferred to a local medical treatment facility and later released. I made the appropriate logbook entries; contacted dispatch; and filled out a fumes event form.Time compression coupled with complex QRH procedures significantly increased the workload during this event. SOP's greatly aided in safe outcome of this event. This was the first trip and first leg the FO and I had ever flown together; and strong SOP compliance allowed us to complete every required task effectively. I don't know what occurred mechanically with the aircraft to cause the fumes; so it is difficult to suggest any preventative measures.