In case your CEO and board decided to decrease your marketing budget claiming that business is running smoothly and they would direct it to other needed investment. How would you persuade them not to and what would be your alternatives if they insisted on ?

Ultimately it is their decision to make.  But you need to show them the relationship between marketing budget and turnover, and what effect reducing the marketing budget is likely to have.

I would remind the board that Marketing is a necessary element in the discovery of consumer needs and tendencies and that the competition is more fierce then ever in establishing brand awareness and retention of customers.  Marketing is not just advertising and promoting a product/service.  It also entails developing and maintaining customer relationships through which prospective buyers are identified, understood intimately, and influenced to embrace a favorable long-term perception of the organization and what it offers so that they will choose you among all others in the marketplace.

If this is not compelling enough to change the mind of the board, then I would have to suggest delaying R&D on new product development and focus the limited funds on continuing the promotion of existing products/services to a least maintain marketshare and brand loyalty.