Yes and no. It's a complex question to ask.

Theoretically, if people started saving money en masse then the economy would take a hit in the short term as all that cash leaves the system temporarily. But when people save money, they tend to do it with a purpose. Either to build an emergency fund, or to make a large purchase. So eventually that money does find it's way back, one way or another. And if someone does manage to save enough to do something exceptional with it, such as start a business or buy stock, then that too is good for the economy.

In one sense, it shouldn't matter which way you go with it if all you're concerned with is immediate economic health. People saving money or people spending money, it should all balance out in the long run.

The issue with people spending money and not saving, however, is that people with little savings are more susceptible to economic downturns. When jobs are lost and prices rise people are forced to take out loans, run up credit cards, and forego opportunities for advancement such as taking college classes. This sets everybody back, and if people can't pay back their debts before the next downturn, then they will end up in an ever deepening pit of debt, and the creditors will not see a return on their investment, hurting the economy even further as it becomes harder to make new investments.

That last paragraph is why the recovery from 2008 has been so slow. People are still in debt, nobody has savings. Most Americans live paycheck to paycheck, even those in the higher brackets. And debt of all kinds increased during and after the downturn, and people are having trouble paying it back with wages and living expenses being what they are.