Yes, it is the government’s responsibility to do what it can to balance out these cycles. However, it is very hard for the government to do so without making mistakes. Therefore, the government should try harder to do what is economically right.
It would be a bad idea to simply allow the economy to try to fix itself in times of recession. It may be true that the economy will self-correct in all instances. However, it will only do so gradually. There may be a long interval in which the economy struggles as it tries to fix itself. The government should not allow people to suffer through such intervals if it has the means to help them.
That said, the government does not always do a good job of trying to balance out the business cycle. It often does the wrong things. Therefore, the government should be at least a little bit “humble” and not try to do more than it really is able to.
However, regardless of whether it always does well, the government has the responsibility to try to make sure that its people do not suffer when it could do something to prevent that suffering.