How do you effectively communicate the importance of investing in company culture to the Board?

Previous answers both excellent.  It's also worth understanding who your board is - you may have an open door and can push their thinking, get their input, etc; or you may have a Board who just aren't interested in 'all that people stuff', in which case remedial is where it's at and aiming for acknowledgement of the impact of culture is a good start!  Hopefully more of the first and none of the second :D

There are two stories you need to tell because different people will need different stories.  
1. The Story for the Risk Averse - This story is a story of Breaking Bad.   What is the RISK OF NOT INVESTING in culture.  What is the worst case and likely case scenario of not adapting your systems, of not listening to employees, and such.  And in particular if your competitors are investing in culture.    They both reduce costs, improve quality, and are able to expand long term while you met a quarterly finacial target.    The Risk of Not investing in holistic culture change which goes much deeper than posters on the wall is too high to risk.    https://medium.com/@charlierataj/how-to-both-reduce-costs-and-unlock-your-organizations-potential-dc9bfecd27ff

2. The Story for the Opportunity Focused:  This is less about the bad things that happen by not investing, but the way that investiing in your culture sets you apart in the market, and that by being intentional about the values which differentiate your organization in the market place means that culture is not just some fluffy thing, but a strategic imperative that is not a cost, but a value add to the customer.    

The other story to tell may be about the journey...  This can feel really heavy, but doesn't have to be sometjhing that happens all at once.   It can start with some well designed experiments in parts of the organiation, or better yet with the board and exective team themselves.  (how they meet, who decides what).    BUilding a little bit of infrastructure so that investing in culture is a routine habit and not a surgery is a big deal and has a great deal more staying power.  

You have to find KPIs the board cares about and then either directly or indirectly connect the dots between the impact culture has on those metrics. Then find sources they respect (ex. Gallup).  I would also position whatever you do as an investment with a tangible ROI in the form or moving in the right directions against those metrics on a time horizon (some KPIs will take longer to impact).