In the country where I live (Switzerland), the state is not strictly secular, and is supporting some churches. The vast majority of the population is not religious anymore (regardless of whether or not they're believers), and only a small fraction of people attend (and thus make donations) to their church regularly. I do not know the details of their financing but I'm pretty confident the Churches depend largely on state subventions and would have to drastically reduce their offers and activities if the state's financial support were to disappear.
In France however, the world's first country to adopt secularism in 1789, and thus the direct inspiration for many other secular states, the system works as follows : The state does not recognize nor support any religious organisation.
I can imagine that the majority church (in this case, Roman Catholics) still have barely enough believers to survive uniquely based on donations, but how do minority churches, such as Protestants, survive ? Do believers have to give really huge donations in order to not see their Church collapse ?