(CNN) -- A 1-year-old girl who became seriously ill on a sailboat hundreds of miles off Mexico was on a U.S. Navy frigate Sunday and in stable condition, the U.S. Coast Guard said.
Her parents thanked those who played a part in their rescue.
"We are very thankful to be safe and well. We also appreciate all the concern, thoughts and prayers of everyone back home for the health of our daughter Lyra. She is doing well now, and her medical condition continues to improve," Eric and Charlotte Kaufman said in a statement.
They defended themselves against critics who question their decision to sail with their family.
"Please know that this is how our family has lived for seven years, and when we departed on this journey more than a year ago, we were then and remain today confident that we prepared as well as any sailing crew could," they wrote.
"The ocean is one of the greatest forces of nature, and it always has the potential to overcome those who live on or near it.
"We are proud of our choices and our preparation, and while we are disappointed that we lost our sailboat and our home, we remain grateful for those who came to our aid and those family and friends who continue to encourage and support us."
On Thursday morning, the Coast Guard received a distress call about the girl, who was on the Rebel Heart -- a Hans Christian 36 boat -- with her parents and sister about 900 miles from Cabo San Lucas, Mexico.
The sailboat did not have steering or communication abilities.
Members of an Air National Guard unit operating out of Moffett Federal Airfield in the San Francisco Bay Area were dispatched, according to a spokesman. Four members of the 129th Rescue Wing parachuted into the ocean, inflated a motorized boat and reached the 36-foot sailboat, said 2nd Lt. Roderick B. Bersamina.
The crew of the USS Vandegrift picked up the family and the four rescuers.
"Flight doctor says baby is stable after transfer; aircraft & air crews in Cabo are prepared,alert & evaluating all options," the 129th Rescue Wing wrote on its Twitter page Sunday.
Officials said they can't give details about the illness, but said "the family is in good spirits."
The family will stay aboard the Vandegrift as it sails back to San Diego.
They are expected to arrive either Wednesday or Thursday, according to 129th Rescue Wing Group Commander Col. Hernando Polo.
Maj. Mark Bomann said the baby is stable.
"We've been quite pleased with her status. The family is exhausted, as one can imagine, but they're out of any imminent danger or harm, which is what you'd worry about in a boat that may have problems," he told reporters.
According to their blogs, Eric Kaufman and his wife, Charlotte, had plans to cross the Pacific and left Mexico some two weeks ago with their daughters Cora, 3, and 1-year-old Lyra. The family lives in San Diego.
They left there in 2012 and were slowly making their way around the world. In one of her earlier blog posts, dated October 2012, Charlotte said that she and her husband had done a ton of research and would never purposefully put their family in harm's way. She wrote that they traveled with a satellite phone.
"Also remember that it is far, far more dangerous to drive every day on the freeway than it is to sail from San Diego to Mexico, or even around the world," Charlotte wrote.
CNN's Janet DiGiacomo, John Newsome and AnneClaire Stapleton contributed to this story