Mr Alexander said one of the challenges would be to "bring Scotland together" in three weeks' time following the historic vote.
The leading Labour politician said he had been called "scum", a "quisling" and "Judas" after speaking out in favour of the Union earlier this week.
And a caller on BBC Radio Scotland's Morning Call programme today also attacked him for "scaremongering" over independence, before accusing the Paisley and Renfrewshire South MP of being a "f****** liar".
The man, identified only as Tom, told Mr Alexander: "Your party has been continually fooling us for years, and I don't appreciate your scare tactics.
"I would like a straight answer - something you've not been able to give in nearly an hour."
He pressed Mr Alexander on the issue of currency in an independent Scotland, saying claims the country would not be able to use the pound were "about scaremongering from your people".
Presenter Louise White intervened, saying: "Tom, I've been told by my research team that you've got a specific question about student fees. Do you want to put that question to Mr Alexander?"
But the caller then responded: "There's not much point, he's just a f****** liar."
Mr Alexander later told the programme: "One of our challenges after September 18 is to bring Scotland together. This referendum is both energising Scotland and it is dividing Scotland."
He said that within 20 minutes of being interviewed on radio on Monday morning, "on social media I was being called a quisling, I was being called Judas, I was being called scum".
He stressed: "I think we need to work very hard as a community and a nation to leave that behind, because we're going to have to work together for a better future for Scotland, whatever the outcome on September 19.
"I think our most immediate task will be to bring Scotland together. That's why I welcome the initiative that has been taken by the Moderator of the Church of Scotland in saying he wants to convene a national service of reconciliation at St Giles Cathedral (in Edinburgh), because this debate has divided Scotland and I think we're going to have to come together in three weeks' time."
Mr Alexander also used the programme to condemn Scottish Government threats that an independent Scotland would not take on a share of the UK national debt if no deal over a currency union could be reached.
In a television debate earlier this week, Scottish First Minister Alex Salmond repeated his threat that if the rest of the UK denied Scotland the assets of the Bank of England, Scotland would not accept the liabilities of its share of the debt - and said there is ''no way'' a UK chancellor would allow that to happen.
Mr Alexander said: "I was genuinely surprised on Monday evening when Alex Salmond confirmed very explicitly the threat of defaulting on our debts.
"As Scots I think we take seriously the idea of meeting our obligations and I was genuinely surprised that as a Scot he felt it was appropriate to raise the spectre of walking away from debts that have in part funded services here in Scotland, as well as across the rest of the United Kingdom.
"It's an incredibly dangerous position to take for a man in government - it's a bit like saying 'after we divorce I'm going to hold on to the joint credit card and if you don't give me the joint credit card after the divorce I'm going to burn the house down'.
"It is very, very high stakes, and actually what happens in circumstances like that is the international markets close down, they judge that you are not safe to have lending. We all know from our own personal experience if you don't pay a debt it doesn't go away, it just trashes your credit rating."