This graph from Bank of England bulletin - shows how the private sector saving ratio increased rapidly during the recession of 2008-09 (as people sought to pay off debts). But, due to government spending cuts and declining real wages, the saving ratio is forecast to plummet. (Image Source: Bank of England)
Often when we talk about borrowing we focus on one aspect. Let us say government borrowing. But for a fuller picture, it is worth looking at net debt levels i.e. consider both public sector and private sector debt. Often an increase in government debt is a response to falling private sector debt (rise in savings and fall in spending)
Similarly a drastic cut in government debt, can lead households to respond by running down savings and increasing their borrowing to deal with the austerity measures.
In a recession should a government borrow?
In a recession, people respond to the threat of recession by saving more. This is known as the paradox of thrift. - Because everyone saves, the economy contracts (recession). The justification for government borrowing is to create demand because the private sector is withdrawing. In this case, government borrowing doesn't cause crowding out because the government are helping unemployed resources to be used.
(In normal periods of economic growth, government borrowing would cause crowding out (i.e. government spending increases at expense of private sector spending)
The effect is that although government borrowing rises, private sector borrowing falls. - People seek to pay off their debt.
Therefore, although the rise in public sector debt looks bad. It should be remembered private sector debt was falling. In the US the period after the great recession led to a fall in the growth of net debt. See: US Debt levels
In the UK, the government are cutting spending drastically to try and reduce public sector debt. The result is that UK consumers are responding by cutting saving and increasing borrowing. Falling living standards are causing people to run down their savings.
Therefore, to some extent the reduction in government borrowing will lead to an increase in private sector borrowing.
According to Treasury Select committee, household debt is set to rise from £1,560bn in 2010 (160% of household income) to £2,126bn in 2015 (175% of income). See: Household debt up