The short answer to the question is "no" but there are a few different reasons why. Most people get the first order negative impact of reduced spending on the economy -- there will be fewer people making cars and selling Dippin Dots if there are fewer people buying those things. Typically the counterargument is that when people save they don't typically just hide currency in a pile; at the very least it is put into a bank account and then can be invested in the broader economy in the form of loans, or it is used to buy bonds and finance capital spending, or it's used to buy stocks of companies when they go public. The purchase of existing stocks is a slightly more confusing case because obviously for every buyer there is a seller, which could be someone selling in order to consume or in order to invest elsewhere; regardless, the argument is made that your money is out there doing something and therefore in the scheme of things high savings rates do not drag on the economy.
This is a shortsighted point of view, however. The whole reason passive investment is attractive/necessary in the first place is that central banks work very hard to ensure that there is a slow rate of inflation that eats up savings which are not invested. Otherwise, many more people would just keep their money in piles of physical currency or otherwise park it in the safest-possible vehicles which are the least liquid and the least useful to capital. Monetary policy thereby helps to ensure that savers will supply a sufficient amount of money to capital and as long as that quantity doesn't outpace the development of the productive forces it will recoup a real return.
But when savings do outpace the rise in labor productivity, the rate of labor exploitation, and/or the absolute size of the capitalist economy, several things can happen. The first is that the flood of money into investment vehicles causes their prices to rise and yields to fall; this is why Japanese 10-year government bonds yield only .6% even though Japanese government debt is self-evidently riskier than many other sovereign debts. The second is that a sustained decline in consumption can set off a deflationary spiral which central banks find it very difficult to escape (a deflationary spiral occurs when low consumption causes prices to fall, which encourages even lower consumption as savers realize that they can grow their buying power by just sitting on cash and waiting for it to increase in value). Third, the vague notion that savings will be turned into new loans depends on there existing consumers who want to borrow money, which is dubious in a hypothetical frugaltopia. In fact, a lack of demand for loans is also a giant obstacle to the present Japanese economy. In their case, the problem is principally demographic. As people get old they tend to have little use for loans on any terms because they are no longer interested in buying bigger houses or more cars -- i.e. they have become "frugal." So no matter how much stimulus money the Japanese government throws at markets or how attractive repayment terms are the so-called multiplier effect barely even registers.
Another example worth considering is the present no-holds-barred effort by the Chinese government to increase domestic consumption. China is not an old country demographically but like Japan savings rates there are extremely high -- as high as 30 or 40% in a country that is still quite poor. In the Chinese case, the principal drivers of savings are (1) housing prices which have inflated astronomically and demand enormous down payments, (2) a shabby healthcare system with high out of pocket costs, and (3) legal and social obligation on the part of children to provide for their elderly parents. So Chinese end up just heaping enormous mountains of RMB in the bank, where it sits earning below-inflation interest rates, because that's their only option. That's bad in the first place because it conflicts with national policy to transition to a consumer-oriented middle-income economy (and the failure of national policy in this area is domestically destabilizing), and in the second because China actually has the opposite loan problem that Japan does. Demand for loans is very high and so Chinese banks are just taking the money and throwing it all over everywhere giving any yahoo a loan to do any crazy old speculative thing, which in practice is further driving up housing prices and exacerbating the problem.
So under capitalist conditions, frugality (whether by informed choice or simply as a matter of biological aging) is economically disastrous.
