This post goes over a simple discussion (based on principles concepts) of the act of saving and its affect on an economy's long run growth. The short answer is that yes, higher savings rates are good for long run growth. However, there are a few caveats. If we begin with the simple assumption that:
Y = C + I + G
We find the first argument against saving because it lowers C (consumption) which reduces Y (Gross Domestic Product). We hear this argument on the news often, because of the assumption that our economy is driven by consumer spending. However, if we add another assumption:
I = S
This supposes that I (investment) in the economy is going to be equal to S (savings), so the reduction in consumer spending is going to be 100% offset by the increase in investment spending with no impact on GDP.
However, this is a static equation. If we explore the Solow model which introduces growth, we can explore the nuance in saving a little deeper.
If you explore the Solow growth model in depth, you find what is called steady state condition where the amount saved (and therefore invested in capital) must be equal to the amount of capital lost due to depreciation (or if we care about well being and not just the economy, population growth as well).
If we imagine a factory, we know that things within the factory will break down over time and will need to be replaced. This is called depreciation. In order to acquire the money to pay for this maintenance we need investment which comes from savings. If our savings are inadequate to cover depreciation we will see less factories (capital) over time which reduces productivity in the economy.
However, if the amount we save is larger than depreciation we will see new or larger factories (more capital) form. This is considered good for the economy because it increases productivity.
No matter the savings rate, more savings will translate into more capital. However, more capital means that we will have more depreciation and need more savings to maintain the higher capital levels. This might not sound like a big deal, but remember that everyone has to choose between saving and consuming.
So saving more is good for long run growth, but it comes at the cost of current consumption. Most people enjoy consuming things (especially food, housing, smart phones, etc.) and it can be hard to motivate people to curb consumption to boost savings. However, if savings become small enough, we will actually see depreciation overtake savings and the economy may shrink (assuming population growth is 0).