Following the rate cut Paul Broadhead of the Building Societies Association (BSA) told the BBC that savers were being "punished", arguing that the move could hinder the funds available to societies to lend as mortgages.
Meanwhile, many lenders responded by cutting some of their mortgage rates.
Halifax said it would pass on the rate cut to customers with standard variable rate mortgages.
Other lenders to pass on the rate change to holders of such mortgages include Nationwide, Barclays, Lloyds TSB and Skipton Building Society.
Some business groups said they were not sure further rate cuts would work.
The Federation of Small Businesses (FSB) said that what was needed was improved access to capital.
For and against: Tim Ross, on a tracker mortgage and saver Gloria Harris
It said that a survey of its members found that 63% wanted rates to remain at their current level, compared with only 24% who wanted a further cut.
"These figures suggest that the recent interest rate cuts are not having the desired effect and other means of economic stimulus are required," said FSB national chairman John Wright.
However, the Bank of England argues that "although the transmission mechanism of monetary policy was impaired, the past cuts in Bank Rate would in due course nevertheless have a significant impact".
Others business groups welcomed the cut. The Institute of Directors' chief economist Graeme Leach said: "The interest rate transmission mechanism is clearly impaired but it is not yet kaput".
The Ernst & Young Item Club supported the Bank's decision, but added that the economy was in "deep recession" and believed that interest rates should drop further - "possibly to zero".
Hetal Mehta, senior economic adviser to the Ernst & Young Item Club, highlighted the "difficult task" the Bank faced.
"Six months ago the Bank was balancing slowing economic growth with accelerating inflation.
"However the Bank now has to act to avoid deflation without fear of a further weakening of sterling; a weaker currency should serve to add to the competitiveness of exports."
Responding to Thursday's cut, John Philpott, chief economist at the Chartered Institute of Personnel and Development said: "The Monetary Policy Committee is right to cut Bank Rate to 1%, even though some question the merit of doing so without greater effort to increase the availability of credit to hard-pressed businesses."
But he added: "With conditions in the job market deteriorating rapidly what's needed now to stem the rise in unemployment is early action to boost the supply of money to our cash-strapped economy."
In contrast to the Bank of England the European Central Bank kept eurozone rates on hold at 2% on Thursday.
The rate cut comes against a backdrop of gloomy economic data.
The economy contracted by 0.6% between July and September, and by 1.5% from October to December, Office for National Statistics (ONS) figures showed.
And the ONS also said UK unemployment had risen to 1.92 million in the three months to November 2008 - the highest level since 1997.
Figures from the Purchasing Managers' Index (PMI) released earlier this week showed manufacturing remained weak last month, despite a slight improvement on December.
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