How many banks does a country typically need relative to its population size?

The number of Banks any given country economy can have is not determined by any preset formula (especially with respect to the population). However, the central bank of every country can set minimum limits on the number of branches a bank is supposed to operate (depending on their banking license). For the developing world, many central banks have a requirement to cover the rural areas, so as to expand their banking network/outreach to places not previously covered.

Community banks for example, operate on an entirely different premise, and are more region specific.

The whole goal of the central bank is to ensure that regular banking needs are fulfilled for both the urban population and rural population. With the advent and advances in mobile communication and better communication networks, branchless banking is taking many countries by the storm. No more need to setup expensive physical infrastructure, when a mobile phone can do all your non-cash and even cash simulated banking transactions.

The monopoly control authority in each country would ensure there are enough banks to ensure that no one bank has a monopoly or a few banks to have an oligopoly in the economy that they operate out of.