In the online statement, his mother, Paula Somers, says: "Please... give us an opportunity to see our Luke again."
A man identifying himself as Mr Somers, who was abducted in 2013, appeared in a separate video on Wednesday, saying his life was in danger and asking for help.
The US has revealed it tried to rescue him last month.
"Regrettably, Luke was not present, though hostages of other nationalities were present and were rescued," the National Security Council said on Thursday.
In a video posted on YouTube, Mr Somers' mother and brother said he was "only trying to do good things for the Yemeni population".
"Luke is only a photojournalist and is not responsible for any actions the US government has taken," his brother, Jordan, said.
Noting that her son "appears healthy" in his captors' video, Paula Somers said: "We thank you for that."Mr Somers, 33, worked as a journalist and photographer for local news organisations and his material appeared on international news outlets, including the BBC News website.
In the video released on Wednesday, a member of al-Qaeda in the Arab Peninsula (AQAP) threatens to kill him unless unspecified demands are met.
In a statement afterwards, the White House said President Barack Obama had authorised a rescue operation to free Mr Somers and other hostages last month, but that Mr Somers was not present at the time of the raid.
Classified
On 25 November, US and Yemeni forces rescued six Yemenis, a Saudi and an Ethiopian being held by AQAP in an operation at a mountain cave in the remote Hajr al-Sayar district of Hadramawt province. Seven militants were reportedly killed.
US National Security Council spokeswoman Bernadette Meehan said: "As soon as the US government had reliable intelligence and an operational plan, the president authorised the department of defence to conduct an operation to recover Mr Somers."She added: "The details of the operation remain classified.
"The overriding concern for Mr Somers' safety and the safety of the US forces who undertake these missions made it imperative that we not disclose information related to Mr Somers' captivity and the attempted rescue."
Ms Meehan said the president "could not be prouder" of the US forces who carried out the mission.
AQAP's threat to kill Mr Somers follows the murder of five Western hostages - including three Americans - since August by the Islamist militant group Islamic State, which controls parts of Syria and Iraq.
Scrutiny of US policy on dealing with kidnappers has increased following the killings, reports the BBC's Aleem Maqbool in Washington.
The Obama administration has been criticised for not paying ransoms, not allowing hostage families to speak out and not taking opportunities to negotiate.
While the White House stands by its policies, the president has ordered a review, our correspondent says.
AQAP is regarded by the US as one of the deadliest offshoots of al-Qaeda.
The group is based in eastern Yemen and has built up support amid the unrest which has beset the impoverished country since the overthrow of President Ali Abdullah Saleh in 2011.
