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Huw Edwards is in hospital with "serious mental health issues", his wife says, as she named him as the BBC presenter at the centre of allegations.
His wife Vicky Flind said she was issuing a statement on his behalf after days of speculation "primarily out of concern for his mental well-being and to protect our children".
The Sun has claimed he paid a young person for sexually explicit images.
The Met Police says Edwards will not face any police action.
The family statement said Mr Edwards intends to respond to the allegations personally when he is well enough.
The statement read: "In light of the recent reporting regarding the 'BBC Presenter' I am making this statement on behalf of my husband Huw Edwards, after what have been five extremely difficult days for our family. I am doing this primarily out of concern for his mental well-being and to protect our children.
"Huw is suffering from serious mental health issues. As is well documented, he has been treated for severe depression in recent years.
"The events of the last few days have greatly worsened matters, he has suffered another serious episode and is now receiving in-patient hospital care where he'll stay for the foreseeable future."
In a statement issued to the Press Association, said she hoped the statement would bring an end to speculation about BBC presenters unconnected to the allegations.
The statement also said Mr Edwards "was first told that there were allegations being made against him last Thursday". It also appealed for privacy.
A separate Met statement said: "Detectives from the Met's Specialist Crime Command have now concluded their assessment and have determined there is no information to indicate that a criminal offence has been committed.
"In reaching this decision, they have spoken to a number of parties including the BBC and the alleged complainant and the alleged complainant's family, both via another police force."
The BBC said it will continue its "fact finding investigations" into the allegations.
A spokesperson for the corporation said: "We have seen the statement from the police confirming they have completed their assessment and are not taking further action. We're grateful to them for completing this work at speed.
"The police had previously asked us to pause our fact finding investigations and we will now move forward with that work, ensuring due process and a thorough assessment of the facts, whilst continuing to be mindful of our duty of care to all involved."