Ed Miliband has said he is "not sorry to see Nigel Farage leaving the political scene".
The former Labour leader said Ukip had a record of "stirring up division" under Farage's leadership.
"I'm not sorry to see Nigel Farage leaving the political scene," he told the BBC's Victoria Derbyshire show.
Nigel Farage has refused to say who he believes should replace him as leader of the UK Independence Party.
When asked by ITV News Political Correspondent Carl Dinnen on who would be best suited to replace him, Mr Farage said: "I'm going to keep very quiet on that, and may the best man or the best woman win."
Farage stood down on Monday morning, saying he "never wanted to be a career politician".
Rob Ford, a politics lecturer and author who wrote a book about Ukip, is among those suggesting Nigel Farage's resignation could benefit Ukip's bid to win seats in Labour and Conservative strongholds.
If UKIP replace Farage with someone less polarising & with broader appeal, then both Lab & Con with have major headache on their hands
Not celebrating Nigel Farage's resignation. UKIP could become a greater threat in northern Labour seats under a new leader.
Senior Ukipper talking about "dream ticket" of leader/deputy to target Labour in north and Tory Leavers in the south.
Nigel Farage has insisted that Britain "must have a Brexit Prime Minister".
"As I see it there are three candidates that supported the Brexit campaign, and I'm not going to damn any one of them by offering them my support at this moment in time.
"This is not a moment for business as usual. This is a moment for radical leadership for inspiration and to take advantage of the unbelievable global opportunities that are there before us," he added.
Brexiteers have "wreaked havoc and then left others "to pick up the pieces", Labour MP Chris Bryant has said after Nigel Farage resigned as Ukip leader.
I feel.so angry that all these brexiteers have wreaked havoc and then left the scene for others to pick up the pieces. Wild irresponsibility
Since Britain voted to leave the European Union on June 23:
Nigel Farage has ruled out making a return into politics.
He said: "I was persuaded to come back for one reason, and that was to fight this referendum. I'm very pleased I changed my mind just over a year ago and came back and fought this referendum.
"I won't be changing my mind again, I can promise you", he added.
MPs have questioned how long it will take Nigel Farage to change his mind about resigning this time.
Farage quit as Ukip leader after failing to win a seat in the 2015 general election - only to famously change his mind days later.
Just to be clear - is it a permanent resignation this time? If I remember the last one lasted 3 days...? https://t.co/JbrJ3PWKFe
BREAKING: Nigel Farage has quit has Party Leader of #UKIP. (Again). (We'll wait to see if he comes back. Again).
Nigel Farage has paid tribute to his supporters after he announced his decision to step down as Ukip leader.
"Delighted with the result we fought for so long to achieve," he said, referring to the Brexit vote.
Nigel Farage's decision to resign appears to have gone down well with the party's only MP Douglas Carswell.
The two frequently clashed over the party's tactics during their time working together.
Nigel Farage has dramatically resigned as leader of Ukip - less than two weeks since Britain voted to leave the European Union.
Farage, 52, has had two stints as leader of the Eurosceptic party since 2006.
In a speech in London on Monday morning, he said: "I feel it's right that I should now stand aside as leader of Ukip.
"I will continue to support the party, I will support the new leader, I will watch the renegotiation process in Brussels like a hawk and perhaps comment in the European Parliament from time to time.
"It has been a huge chunk of my life doing this, and it's not easy.
"Perhaps when you feel a degree of ownership of something to let it go but it does come at a cost to me, and to those around me.
"During the referendum I said I want my country back. What I'm saying today, is I want my life back, and it begins right now."