First of all, it is important to note that we in the United States are not, so far as we know, in a recession today. The deep recession began in December of 2007 and lasted until June of 2009. It is true that our recovery has not been rapid, but we are not in a recession. With regard to your question, however, there have been both classical and Keynesian policy solutions suggested. For the most part, the policies that have actually been adopted have been Keynesian.
Keynesians would call for tax cuts and/or increases in spending during a recession. We have seen that this has, in fact, been done by the government. The “Bush tax cuts” have been extended. There have been cuts in payroll taxes. There has been an increase in government spending. The most prominent example of this was the stimulus package that was passed by Congress soon after President Obama took office. By doing these things, the government was trying to put more money into the hands of consumers, just as Keynesians would prescribe.
There are those who have proposed more classical solutions. The classical solution to a recession is for the government to back away and simply let the economy fix itself. Therefore, proposals (mainly by Republicans) to reduce the role of the government in the economy are more like classical proposals. When people propose reducing government regulation of business, they are proposing a classical policy. When they propose reforming the tax code to make it more neutral, they are proposing classical policies. However, these policies have largely not been adopted. Most of what has been done is more Keynesian in nature.