Wow - it would take me a long time to find the appropriate sources to answer this the way I want to. I also feel obliged to admit that I have an opinion on this topic, and I hope that @Samuel Russell or someone from the other side of the aisle will contribute to counterbalance me.
@John Craven is correct that the great depression was a complex, global phenomenon, and Roosevelt's policies are only one factor. I'll agree with Mr. Craven that it would be absurd to credit Roosevelt with sole agency in either direction.
With that disclaimer out of the way, I'll once again reference Niall Ferguson's Ascent of Money. Ferguson points out that the stock market returned to reasonable levels rather quickly. The driver of the depression wasn't the stock market crash, but the US government policy of minimizing liquidity. Banks couldn't loan money to start a business that would employ people. And US Government policy was directly responsible for that. The government feared bank failure more than they feared continued depression, and they required that banks build up and maintain reserves of currency. Banks were unable to lend money, which prolonged the depression.
Contrary to popular opinion, the Great Depression was not caused by the stock market crash of 1929. Rather the consensus among economists is what made the Depression great in the U.S. were the mistakes made by the Fed—especially by its decision, in 1931 and 1932, to allow so many banks to fail. The Fed in 1931 stuck to the principle that it should only lend to institutions facing a temporary shortage of liquidity and that propping up insolvent banks would be throwing good money after bad.
Liquidity shortages destroyed the international monetary system in 1931.
Bank of International Settlement
Milton Friedman and Anna Schwartz argued that steady withdrawals from banks by nervous depositors ("hoarding") were inspired by news of the fall 1930 bank runs and forced banks to liquidate loans, which directly caused a decrease in the money supply, shrinking the economy.
If the US government had eased monetary policy and supported the banks, they could have reduced the fear of bank runs. Hoarding reduces the money supply; investment strengthens the money supply. Of course the US government couldn't ease monetary policy because they were committed to the gold standard.
Roosevelt's committment to the Gold Standard, which prevented the USA from monetarist tools that could have acted as an engine for growth. Update: @pugsville asks whether Roosevelt took us off the gold standard. Roosevelt took us off the international gold standard. The value of the dollar was still tied to gold, but the value was set based on Roosevelt's whim. The international gold standard was still being defended by the Bank of England. Daniel Drezner in his Foundations of Economic Prosperity points out that Roosevelt took the US off the gold standard without coordination with other countries. Had he coordinated our exit, he could have mitigated the damage to the US and world economic system.
The US government took further actions to prolong the depression.
Economists agree that the Smoot–Hawley Tariff Act increased the severity of the Great Depression.1
The mind boggglingly short sighted Smoot Hawley act inhibited trade with other nations, further inhibiting the ability of US industry to recover from the depression. Smoot-Hawley vies with the gold standard for the title of "Stupidest economic policy of all time!".
Roosevelt adopted the idea of raising the price of gold to inflate the currency and reverse the debilitating deflation of prices. The idea came from Professor George Warren of Cornell University. When Roosevelt told Morgenthau he was thinking of raising the price of gold by 21 cents, his entourage asked him why. "It's a lucky number", Roosevelt said. "Because it's three times seven." As Morgenthau later wrote, "If anybody knew how we really set the gold price through a combination of lucky numbers, etc., I think they would be frightened."
Wikipedia citing Shlaes
The US government pursued other policies that numbed the economy and the modern mind. For example, prosecuting illiterate immigrants for the crime of selecting their own chickens rather than accepting the chicken that the government had decided they should receive. Taking that case to the supreme court qualifies as a microeconmic policy which is guaranteed to arrest growth. No sane individual is going to want to take out a loan to start a business if they can't permit the customer to choose their goods.
Did the US government prolong the depression through primitive, bizarre and uninformed macroeconomic policies? Yes, absolutely. Having said that, remember that John Maynard Keynes hadn't yet written the seminal work on macroeconomic policy. And no matter what flaws I ascribe to Roosevelt, the opposition didn't have a solution that worked any better.
The US government, like all governments, assumed that the future would be dominated by planned, centralized economies. They assumed that central government control of demand was the only resolution to any macroeconomic problem. And their assumption seemed to be true. The New Deal was a demand stimulus, and the ultimate resolution to the economic economic crisis was driven by centrally planned demand (a war). But I align with those modern economists who believe that the crisis would have been shorter and less severe if the government had employed monetary policy, eased liquidity, and stimulated growth, rather than avoiding monetary policy, restricting liquidity, and intimidating growth.
As I said at the beginning, I'm not trying to hide my bias; the best you can hope is that Mr. Russell will provide an opposing viewpoint.